Abstract Algebra

posted: 01-Aug-2025 & updated: 03-Aug-2025

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Introduction

Preamble

Notations

• sets of numbers
  • $\naturals$ - set of natural numbers
  • $\integers$ - set of integers
  • $\integers_+$ - set of nonnegative integers
  • $\rationals$ - set of rational numbers
  • $\reals$ - set of real numbers
  • $\preals$ - set of nonnegative real numbers
  • $\ppreals$ - set of positive real numbers
  • $\complexes$ - set of complex numbers
• sequences $\seq{x_i}$ and the like
  • finite $\seq{x_i}_{i=1}^n$, infinite $\seq{x_i}_{i=1}^\infty$ - use $\seq{x_i}$ whenever unambiguously understood
  • similarly for other operations, e.g., $\sum x_i$, $\prod x_i$, $\cup A_i$, $\cap A_i$, $\bigtimes A_i$
  • similarly for integrals, e.g., $\int f$ for $\int_{-\infty}^\infty f$
• sets
  • $\compl{A}$ - complement of $A$
  • $A\sim B$ - $A\cap \compl{B}$
  • $A\Delta B$ - $(A\cap \compl{B}) \cup (\compl{A} \cap B)$
  • $\powerset(A)$ - set of all subsets of $A$
• sets in metric vector spaces
  • $\closure{A}$ - closure of set $A$
  • $\interior{A}$ - interior of set $A$
  • $\relint A$ - relative interior of set $A$
  • $\boundary A$ - boundary of set $A$
• set algebra
  • $\sigma(\subsetset{A})$ - $\sigma$-algebra generated by $\subsetset{A}$, i.e., smallest $\sigma$-algebra containing $\subsetset{A}$
• norms in $\reals^n$
  • $\|x\|_p$ ($p\geq1$) - $p$-norm of $x\in\reals^n$, i.e., $(|x_1|^p + \cdots + |x_n|^p)^{1/p}$
  • e.g., $\|x\|_2$ - Euclidean norm
• matrices and vectors
  • $a_{i}$ - $i$-th entry of vector $a$
  • $A_{ij}$ - entry of matrix $A$ at position $(i,j)$, i.e., entry in $i$-th row and $j$-th column
  • $\Tr(A)$ - trace of $A \in\reals^{n\times n}$, i.e., $A_{1,1}+ \cdots + A_{n,n}$
• symmetric, positive definite, and positive semi-definite matrices
  • $\symset{n}\subset \reals^{n\times n}$ - set of symmetric matrices
  • $\possemidefset{n}\subset \symset{n}$ - set of positive semi-definite matrices; $A\succeq0 \Leftrightarrow A \in \possemidefset{n}$
  • $\posdefset{n}\subset \symset{n}$ - set of positive definite matrices; $A\succ0 \Leftrightarrow A \in \posdefset{n}$
• sometimes, use Python script-like notations (with serious abuse of mathematical notations)
  • use $f:\reals\to\reals$ as if it were $f:\reals^n \to \reals^n$, e.g., $$ \exp(x) = (\exp(x_1), \ldots, \exp(x_n)) \quad \mbox{for } x\in\reals^n $$ and $$ \log(x) = (\log(x_1), \ldots, \log(x_n)) \quad \mbox{for } x\in\ppreals^n $$ which corresponds to Python code numpy.exp(x) or numpy.log(x) where x is instance of numpy.ndarray, i.e., numpy array
  • use $\sum x$ to mean $\ones^T x$ for $x\in\reals^n$, i.e. $$ \sum x = x_1 + \cdots + x_n $$ which corresponds to Python code x.sum() where x is numpy array
  • use $x/y$ for $x,y\in\reals^n$ to mean $$ \rowvecthree{x_1/y_1}{\cdots}{x_n/y_n}^T $$ which corresponds to Python code x / y where x and y are $1$-d numpy arrays
  • use $X/Y$ for $X,Y\in\reals^{m\times n}$ to mean $$ \begin{my-matrix}{cccc} X_{1,1}/Y_{1,1} & X_{1,2}/Y_{1,2} & \cdots & X_{1,n}/Y_{1,n} \\ X_{2,1}/Y_{2,1} & X_{2,2}/Y_{2,2} & \cdots & X_{2,n}/Y_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ X_{m,1}/Y_{m,1} & X_{m,2}/Y_{m,2} & \cdots & X_{m,n}/Y_{m,n} \end{my-matrix} $$ which corresponds to Python code X / Y where X and Y are $2$-d numpy arrays

Some definitions

statement $P_n$, said to happen infinitely often or i.o. if $$ \left( \forall N\in\naturals \right) \left( \exists n > N \right) \left( P_n \right) $$

statement $P(x)$, said to happen almost everywhere or a.e. or almost surely or a.s. (depending on context) associated with measure space $\meas{X}{\algB}{\mu}$ if $$ \mu \set{x}{P(x)} = 1 $$ or equivalently $$ \mu \set{x}{\sim P(x)} = 0 $$

Some conventions

• (for some subjects) use following conventions
  • $0\cdot \infty = \infty \cdot 0 = 0$
  • $(\forall x\in\ppreals)(x\cdot \infty = \infty \cdot x = \infty)$
  • $\infty \cdot \infty = \infty$

Abstract Algebra

Why Abstract Algebra?

Why abstract algebra?

• it's fun!
• can understand instrict structures of algebraic objects
• allow us to solve extremely practical problems (depending on your definition of practicality)
  • e.g., can prove why root formulas for polynomials of order $n\geq 5$ do not exist
• prepare us for pursuing further math topics such as
  • differential geometry
  • algebraic geometry
  • analysis
  • representation theory
  • algebraic number theory

Some history

• by the way, historically, often the case that application of an idea presented before extracting and presenting the idea on its own right
• e.g., Galois used “quotient group'' only implicitly in his 1830's investigation, and it had to wait until 1889 to be explicitly presented as “abstract quotient group'' by Hölder

Groups

Monoids

mapping $S\times S \to S$ for set $S$, called law of composition (of $S$ to itself)

• when $(\forall x, y, z \in S)((xy)z = x(yz))$, composition is said to be associative
• $e\in S$ such that $(\forall x\in S)(ex = xe = x)$, called unit element - always unique for any two unit elements $e$ and $f$, $e = ef = f,$ hence, $e=f$

set $M$ with composition which is associative and having unit element, called monoid (so in particular, $M$ is not empty)

• monoid $M$ with $\left( \forall x, y \in M \right) \left( xy = yx \right)$, called commutative or abelian monoid
• subset $H\subset M$ which has the unit element $e$ and is itself monoid, called submonoid

Groups

monoid $G$ with $$ \left( \forall x \in G \right) \left( \exists y \in G \right) \left( xy = yx = e \right) $$ called group

• for $x\in G$, $y\in G$ with $xy=yx=e$, called inverse of $x$
• group derived from commutative monoid, called abelian group or commutative group
• group $G$ with $|G|<\infty$, called finite group
• (similarly as submonoid) $H\subset G$ that has unit element and is itself group, called subgroup
• subgroup consisting only of unit element, called trivial

Cyclic groups, generators, and direct products

group $G$ with $$ \left( \exists a\in G \right) \left( \forall x \in G \right) \left( \exists n\in \naturals \right) \left( x = a^n \right) $$ called cyclic group , such $a\in G$ called cyclic generator

for group $G$, $S\subset G$ with $$ \left( \forall x \in G \right) \left( x \mbox{ is arbitrary product of elements or inverse elements of } S \right) $$ called set of generators for $G$, said to generate $G$, denoted by $G=\generates{S}$

for two groups $G_1$ and $G_2$, group $G_1\times G_2$ with $$ \left( \forall (x_1,x_2), (y_1,y_2) \in G_1 \times G_2 \right) \left( (x_1,x_2)(y_1,y_2) = (x_1y_1, x_2,y_2) \in G_1 \times G_2 \right) $$ whose unit element defined by $(e_1,e_2)$ where $e_1$ and $e_2$ are unit elements of $G_1$ and $G_2$ respectively, called direct product of $G_1$ and $G_2$

Homeomorphism and isomorphism

for monoids $M$ and $M'$, mapping $f:M\to M'$ with $f(e)=e'$ $$ \left( x,y \in M \right) \left( f(xy) = f(x)f(y) \right) $$ where $e$ and $e'$ are unit elements of $M$ and $M'$ respectively, called monoid-homeomorphism or simple homeomorphism

• group homeomorphism $f:G\to G'$ is similarly monoid-homeomorphism
• homeomorphism $f:G\to G'$ where exists $g:G\to G'$ such that $f\circ g:G'\to G'$ and $g\circ f:G\to G$ are identity mappings, called isomorphism, sometimes denoted by $G\isomorph G'$
• homeomorphism of $G$ into itself, called endomorphism
• isomorphism of $G$ onto itself, called automorphism

• set of all automorphisms of $G$ is itself group, denoted by \aut{G}

Kernel, image, and embedding of homeomorphism

for group-homeomorphism $f:G\to G'$ where $e'$ is unit element of $G'$, $f^{-1}(\{e'\})$, which is subgroup of $G$, called kernel of $f$, denoted by $\Ker{f}$

homeomorphism $f:G\to G'$ establishing isomorphism between $G$ and $f(G)\subset G'$, called embedding

• for group-homeomorphism $f:G\to G'$, $f(G)\subset G'$ is subgroup of $G'$
• homeomorphism whose kernel is trivial is injective, often denoted by special arrow $$ f:G \injhomeo G' $$
• surjective homeomorphism whose kernel is trivial is isomorphism
• for group $G$, its generators $S$, and another group $G'$, map $f:S\to G'$ has at most one extension to homeomorphism of $G$ into $G'$

Orthogonal subgroups

for group $G$ and two subgroups $H$ and $K\subset G$ with $HK = G$, $H\cap K = \{e\}$, and $\left( x\in H, y\in K \right) \left( xy=yx \right)$, $$ f: H\times K \to G $$ with $(x,y)\mapsto xy$ is isomorphism can generalize to finite number of subgroups, $H_1$, , $H_n$ such that $$ H_1 \cdots H_n = G $$ and $$ H_{k+1} \cap (H_1\cdots H_k) = \{e\} $$ in which case, $G$ is isomorphic to $H_1\cdots H_n$

Cosets of groups

for group $G$ and subgroup $H\subset G$, $aH$ for some $a\in G$, called left coset of $H$ in $G$, and element in $aH$, called coset representation of $aH$ - can define right cosets similarly

for group $G$ and subgroup $H\subset G$,

• for $a\in G$, $x\mapsto ax$ induces bijection of $H$ onto $aH$, hence all left cosets have same cardinality
• $aH \cap bH \neq \emptyset$ for $a,b\in G$ implies $aH=bH$
• hence, $G$ is disjoint union of left cosets of $H$
• same statements can be made for right cosets

number of left cosets of $H$ in $G$, called index of $H$ in $G$, denoted by $(G:H)$ - index of trivial subgroups, called order of $G$, denoted by $(G:1)$

Indices and orders of groups

for group $G$ and two subgroups $H$ and $K\subset G$ with $K\subset H$, $$ (G:H) (H:K) = (G:K) $$ when $K$ is trivial, we have $$ (G:H) (H:1) = (G:1) $$

hence, if $(G:1)<\infty$, both $(G:H)$ and $(H:1)$ divide $(G:1)$

Normal subgroup

subgroup $H\subset G$ of group $G$ with $$ \left( \forall x \in G \right) \left( x H = H x \right) \Leftrightarrow \left( \forall x \in G \right) \left( xHx^{-1}= H \right) $$ called normal subgroup of $G$, in which case

• set of cosets $\set{xH}{x\in G}$ with law of composition defined by $(xH)(yH) = (xy)H,$ forms group with unit element $H$, denoted by $G/H$, called factor group of $G$ by $H$, read $G$ modulo $H$ or $G$ mod $H$
• $x \mapsto xH$ induces homeomorphism of $X$ onto $\set{xH}{x\in G}$, called canonical map , kernel of which is $H$

• kernel of (every) homeomorphism of $G$ is normal subgroups of $G$
• for family of normal subgroups of $G$, $\seq{N_\lambda}$, $\bigcap N_\lambda$ is also normal subgroup
• every subgroup of abelian group is normal
• factor group of abelian group is abelian
• factor group of cyclic group is cyclic

Normalizers and centralizers

for subset $S\subset G$ of group $G$, $$ \set{x\in G}{xSx^{-1} = S} $$ is subgroup, called normalizer of $S$, and also called centralizer of $a$ when $S=\{a\}$ is singletone; $$ \set{x\in G}{(\forall y\in S)(xyx^{-1} = y)} $$ called centralizer of $S$, and centralizer of $G$ itself, called center of $G$

• e.g., $A \mapsto \det A$ of multiplicative group of square matrices in $\reals^{n\times n}$ into $\reals\sim\{0\}$ is homeomorphism, kernel of which called special linear group, and (of course) is normal

Normalizers and congruence

subgroup $H\subset G$ of group $G$ is normal subgroup of its normalizer $N_H$

• subgroup $H\subset G$ of group $G$ is normal subgroup of its normalizer $N_H$
• subgroup $K\subset G$ with $H\subset K$ where $H$ is normal in $K$ is contained in $N_H$
• for subgroup $K\subset N_H$, $KH$ is group and $H$ is normal in $KH$
• normalizer of $H$ is largest subgroup of $G$ in which $H$ is normal

for normal subgroup $H\subset G$ of group $G$, we write $$ x \equiv y \Mod{H} $$ if $xH=yH$, read $x$ and $y$ are congruent modulo $H$ - this notation used mostly for additive groups

Exact sequences of homeomorphisms

below sequence of homeomorphisms with $\Img f = \Ker g$ $$ G' \overset{f}{\longrightarrow} G \overset{g}{\longrightarrow} G'' $$ said to be exact below sequence of homeomorphisms with $\Img f_i = \Ker f_{i+1}$ $$ G_1 \overset{f_1}{\longrightarrow} G_2 \overset{f_2}{\longrightarrow} G_3 \longrightarrow \cdots \overset{f_{n-1}}{\longrightarrow} G_n $$ said to be exact

• for normal subgroup $H\subset G$ of group $G$, sequence $H \overset{j}{\to} G \overset{\varphi}{\to} G/H$ is exact where $j$ is inclusion and $\varphi$
• $0 \overset{}{\to} G' \overset{f}{\to} G \overset{g}{\to} G'' \overset{}{\to} 0$ is exact if and only if $f$ injective, $g$ surjective, and $\Img f = \Ker g$
• if $H=\Ker g$ above, $0 \overset{}{\to} H \overset{}{\to} G \overset{}{\to} G/H \overset{}{\to} 0$
• more precisely, exists commutative diagram as in the figure, in which vertical mappings are isomorphisms and rows are exact

Canonical homeomorphism examples

all homeomorphisms described below called canonical

• for two groups $G$ & $G'$ and homeomorphism $f:G\to G'$ whose kernel is $H$, exists unique homeomorphism $f_*: G/H \to G'$ with $$ f=f_*\circ \varphi $$ where $\varphi:G\to G/H$ is canonical map, and $f_*$ is injective
  • $f_*$ can be defined by $xH\mapsto f(x)$
  • $f_*$ said to be induced by $f$
  • $f_*$ induces isomorphism $\lambda: G/H \to \Img f$
  • below sequence summarizes above statements $$ G \overset{\varphi}{\to} G/H \overset{\lambda}{\to} \Img f \overset{j}{\to} G $$ where $j$ is inclusion
• for group $G$, subgroup $H\subset G$, and homeomorphism $f:G\to G'$ whose kernel contains $H$, intersection of all normal subgroups containing $H$, $N$, which is the smallest normal subgroup containing $H$, is contained in $\Ker f$, i.e., $N\subset \Ker f$, and exists unique homeomorphism, $f_*:G/N\to G'$ such that $$ f = f_* \circ \varphi $$ where $\varphi:G\to G/H$ is canonical map
  • $f_*$ can be defined by $xN\mapsto f(x)$
  • $f_*$ said to be induced by $f$
• for subgroups of $G$, $H$ and $K$ with $K\subset H$, $xK \mapsto xH$ induces homeomorphism of $G/K$ into $G/H$, whose kernel is $\set{xK}{x\in H}$, thus canonical isomorphism $$ (G/K)/(H/K) \isomorph (G/K) $$ this can be shown in the figure where rows are exact
• for subgroup $H\subset G$ and $K\subset G$ with $H$ contained in normalizer of $K$, $H\cap K$ is normal subgroup of $H$, $HK=KH$ is subgroup of $G$, exists surjective homeomorphism $$ H \to HK / K $$ with $x \mapsto xK$, whose kernel is $H\cap K$, hence canonical isomorphism $$ H/(H\cap K) \isomorph HK/K $$
• for group homeomorphism $f:G\to G'$, normal subgroup of $G'$, $H'$, $$ H=f^{-1}(H')\subset G $$ as shown in the figure, $H$ is normal in $G$ and kernel of homeomorphism $$ G \overset{f}{\to} G'\overset{\varphi}{\to} G'/H' $$ is $H$ where $\varphi$ is canonical map, hence we have injective homeomorphism $$ \bar{f}:G/H \to G'/H' $$ again called canonical homeomorphism, giving commutative diagram in the figure; if $f$ is surjective, $\bar{f}$ is isomorphism

Towers

for group $G$, sequence of subgroups $$ G = G_0 \supset G_1 \supset G_2 \supset \cdots \supset G_m $$ called tower of subgroups

• said to be normal if every $G_{i+1}$ is normal in $G_i$
• said to be abelian if normal and every factor group $G_i/G_{i+1}$ is abelian
• said to be cyclic if normal and every factor group $G_i/G_{i+1}$ is cyclic

for group homeomorphism $f:G\to G'$ and normal tower $$ G' = G'_0 \supset G'_1 \supset G'_2 \supset \cdots \supset G'_m $$ tower $$ f^{-1}(G') = f^{-1}(G'_0) \supset f^{-1}(G'_1) \supset f^{-1}(G'_2) \supset \cdots \supset f^{-1}(G'_m) $$ is

• normal if $G'_i$ form normal tower
• abelian if $G'_i$ form abelian tower
• cyclic if $G'_i$ form cyclic tower

because every homeomorphism $$ G_i / G_{i+1} \to G'_i / G'_{i+1} $$ is injective

Refinement of towers and solvability of groups

for tower of subgroups, tower obtained by inserting finite number of subgroups, called refinement of tower

group having an abelian tower whose last element is trivial subgroup, said to be solvable

• abelian tower of finite group admits cyclic refinement
• finite solvable group admits cyclic tower, whose last element is trivial subgroup

group whose order is prime power is solvable

for group $G$ and its normal subgroup $H$, $G$ is solvable if and only if both $H$ and $G/H$ are solvable

Commutators and commutator subgroups

for group $G$, $xyx^{-1}y^{-1}$ for $x,y\in G$, called commutator

subgroup generated by commutators of group $G$, called commutator subgroup, denoted by $G^C$, i.e. $$ G^C = \generates{\set{xyx^{-1}y^{-1}}{x,y\in G}} $$

• $G^C$ is normal in $G$
• $G/G^C$ is commutative
• $G^C$ is contained in kernel of every homeomorphism of $G$ into commutative group
• of above statements

• commutator group is at the heart of solvability and non-solvability problems!

Simple groups

non-trivial group having no normal subgroup other than itself and trivial subgroup, said to be simple

abelian group is simple if and only if cycle of prime order

Butterfly lemma

for subgroups $U$ and $V$ of a group and normal subgroups $u$ and $v$ of $U$ and $V$ respectively, $$ u(U\cap v) \mbox{ is normal in } u(U\cap V) $$ $$ (u\cap V)v \mbox{ is normal in } (U\cap V)v $$ and factor groups are isomorphic, i.e., $$ u(U\cap V) / u(U\cap v) \isomorph\ (U\cap V)v / (u\cap V)v $$ these shown in the figure

• indeed $$ (U\cap V)/((u\cap V)(U\cap v)) \isomorph\ u(U\cap V) / u(U\cap v) \isomorph\ (U\cap V)v / (u\cap V)v $$

Equivalent towers

for two normal towers of same height starting from same group ending with trivial subgroup $$ G = G_1 \supset G_2 \supset G_3 \supset \cdots \supset G_{n+1} = \{e\} $$ $$ G = H_1 \supset H_2 \supset H_3 \supset \cdots \supset H_{n+1} = \{e\} $$ with $$ G_i/G_{i+1}\isomorph H_{\pi(i)+1}/H_{\pi(i)} $$ for some permutation $\pi\in\perm{\{1,\ldots,n\}}$, i.e., sequences of factor groups are same up to isomorphisms and permutation of indices, said to be equivalent

Schreier and Jordan-Hölder theorems

two normal towers starting from same group and ending with trivial subgroup have equivalent refinement

all normal towers starting from same group and ending with trivial subgroup where each factor group is non-trivial and simple are equivalent

Cyclic groups

for group $G$, $n\in\naturals$ with $a^n=e$ for $a\in G$, called exponent of $a$; $n\in\naturals$ with $x^n=e$ for every $x\in G$, called exponent of $G$

for group $G$ and $a\in G$, smallest $n\in\naturals$ with $a^n=e$, called period of $a$

for finite group $G$ of order $n>1$, period of every non-unit element $a$ ($\neq e$) devided $n$; if $n$ is prime number, $G$ is cyclic and period of every generator is $n$

every subgroup of cyclic group is cyclic and image of every homeomorphism of cyclic group is cyclic

Properties of cyclic groups

• infinity cyclic group has exactly two generators; if $a$ is one, $a^{-1}$ is the other
• for cyclic group $G$ of order $n$ and generator $x$, set of generators of $G$ is $$ \set{x^m}{m \mbox{ is relatively prime to }n} $$
• for cyclic group $G$ and two generators $a$ and $b$, exists automorphism of $G$ mapping $a$ onto $b$; conversely, every automorphism maps $a$ to some generator
• for cyclic group $G$ of order $n$ and $d\in\naturals$ dividing $n$, exists unique subgroup of order $d$
• for cyclic groups $G_1$ and $G_2$ of orders $n$ and $m$ respectively with $n$ and $m$ relatively prime, $G_1\times G_2$ is cyclic group
• for non-cyclic finite abelian group $G$, exists subgroup isomorphic to $C\times C$ with $C$ cyclic with prime order

Symmetric groups and permutations

for nonempty set $S$, group $G$ of bijective functions of $S$ onto itself with law of composition being function composition, called symmetric group of $S$, denoted by \perm{S}; elements in $\perm{S}$ called permutations of $S$; element swapping two disjoint elements in $S$ leaving every others left, called transposition

for finite symmetric group $S_n$, exits unique homeomorphism $\epsilon: S_n \to\{-1,1\}$ mapping every transposition, $\tau$, to $-1$, i.e., $\epsilon(\tau)=-1$

element of finite symmetric group $\sigma$ with $\epsilon(\sigma)=1$, called even, element $\sigma$ with $\epsilon(\sigma)=-1$, called odd; kernel of $\epsilon$, called alternating group, denoted by $A_n$

symmetric group $S_n$ with $n\geq 5$ is not solvable

alternating group $A_n$ with $n\geq 5$ is simple

Operations of group on set

for group $G$ and set $S$, homeomorphism $$ \pi:G \to \perm{S} $$ called operation of $G$ on $S$ or action of $G$ on $S$

• $S$, called $G$-set
• denote $\pi(x)$ for $x\in G$ by $\pi_x$, hence homeomorphism denoted by $x\mapsto \pi_x$

• obtain mapping from such operation, $G\times S \to S$, with $(x,s)\mapsto \pi_x(s)$
• often abbreviate $\pi_x(s)$ by $xs$, with which the following two properties satisfied
  • $\left( \forall x,y\in G, s\in S \right) \left( x(ys) = (xy)s \right)$
  • $\left( \forall s\in S \right) \left( es = s \right)$
• conversely, for mapping $G\times S\to S$ with $(x,s)\mapsto xs$ satisfying above two properties, $s\mapsto xs$ is permutation for $x\in G$, hence $\pi_x$ is homeomorphism of $G$ into $\perm{S}$
• thus, operation of $G$ on $S$ can be defined as mapping $S\times G\to S$ satisfying above two properties

Conjugation

for group $G$ and map $\gamma_x:G\to G$ with $\gamma_x(y) = xyx^{-1}$, homeomorphism $$ G \to \aut{G} \mbox{ defined by } x\mapsto \gamma_x $$ called conjugation, which is operation of $G$ on itself

• $\gamma_x$, called inner
• kernel of conjugation is center of $G$
• to avoid confusion, instead of writing $xy$ for $\gamma_x(y)$, write $$ \gamma_x(y) = xyx^{-1} = \prescript{x}{}{y} \mbox{ and } \gamma_{x^{-1}}(y) = x^{-1}yx = {y}^x $$
• for subset $A\subset G$, map $(x,A) \mapsto xAx^{-1}$ is operation of $G$ on set of subsets of $G$
• similarly for subgroups of $G$
• two subsets of $G$, $A$ and $B$ with $B= x A x^{-1}$ for some $x\in G$, said to be conjugate

Translation

operation of $G$ on itself defined by map $$ (x,y) \mapsto xy $$ called translation, denoted by $T_x:G \to G$ with $T_x(y) = xy$

• for subgroup $H\subset G$, $T_x(H) = xH$ is left coset
  • denote set of left cosets also by $G/H$ even if $H$ is not normal
  • denote set of right cosets also by $H\backslash G$
• examples of translation
  • $G=GL(V)$, group of linear automorphism of vector space with field $F$, for which, map $(A,v)\mapsto Av$ for $A\in G$ and $v\in V$ defines operation of $G$ on $V$
    • $G$ is subgroup of group of permutations, $\perm{V}$
  • for $V=F^n$, $G$ is group of nonsingular $n$-by-$n$ matrices

Isotropy

for operation of group $G$ on set $S$ $$ \set{x\in G}{xs = s} $$ called isotropy of $G$, denoted by $G_s$, which is subgroup of $G$

• for conjugation operation of group $G$, $G_s$ is normalizer of $s\in G$
• isotropy groups are conjugate, e.g., for $s,s'\in S$ and $y\in G$ with $ys=s'$, $$ G_{s'} = yG_s y^{-1} $$
• by definition, kernel of operation of $G$ on $S$ is $$ K = \bigcap_{s\in S} G_s \subset G $$
• operation with trivial kernel, said to be faithful
• $s\in G$ with $G_s = G$, called fixed point

Orbits of operation

for operation of group $G$ on set $S$, $\set{xs}{x\in G}$, called orbit of $s$ under $G$, denoted by $Gs$

• for $x,y\in G$ in same coset of $G_s$, $xs = ys$, i.e. $\left( \exists z\in G \right) \left( x,y \in zG_s \right) \Leftrightarrow xs = ys$
• hence, mapping $G/G_s \to S$ with $x \mapsto x G_s$ is morphism of $G$-sets, thus

for group $G$, operating on set $S$ and $s\in S$, order of orbit $Gs$ is equal to index $(G:G_s)$

for subgroup $H$ of group $G$, number of conjugate subgroups to $H$ is index of normalizer of $H$ in $G$

operation with one orbit, said to be transitive

Orbit decomposition and class formula

• orbits are disjoint $$ S = \coprod_{\lambda \in \Lambda} Gs_\lambda $$ where $s_\lambda$ are elements of distinct orbits

for group $G$ operating on set $S$, index set $\Lambda$ whose elements represent distinct orbits $$ |S| = \sum_{\lambda \in \Lambda} (G:G_\lambda) $$

for group $G$ and set $C\subset G$ whose elements represent distinct conjugacy classes $$ (G:1) = \sum_{x\in C} (G:G_x) $$

Sylow subgroups

for prime number $p$, finite group with order $p^n$ for some $n\geq0$, called $p$-group; subgroup $H\subset G$ of finite group $G$ with order $p^n$ for some $n\geq0$, called $p$-subgroup; subgroup of order $p^n$ where $p^n$ is highest power of $p$ dividing order of $G$, called $p$-Sylow subgroup

finite abelian group of order divided by prime number $p$ has subgroup of order $p$

finite group of order divided by prime number $p$ has $p$-Sylow subgroup

for $p$-group $H$, operating on finite set $S$

• number of fixed points of $H$ is congruent to size of $S$ modulo $p$, i.e. $$ \mbox{\# fixed points of }H \equiv |S| \Mod{p} $$
• if $H$ has exaxctly one fixed point, $|S| \equiv 1\Mod{p}$
• if $p$ divides $|S|$, $|S| \equiv 0\Mod{p}$

Sylow subgroups and solvability

finite $p$-group is solvable; if it is non-trivial, it has non-trivial center

for non-trivial $p$-group, exists sequence of subgroups $$ \{e\} = G_0 \subset G_1 \subset G_2 \subset \cdots \subset G_n =G $$ where $G_i$ is normal in $G$ and $G_{i+1}/G_i$ is cyclic group of order $p$

for finite group $G$ and smallest prime number dividing order of $G$ $p$, every subgroup of index $p$ is normal

group of order $pq$ with $p$ and $q$ being distinct prime numbers, is solvable

• now can prove following
  • group of order, $35$, is solvable - implied by and
  • group of order less than $60$ is solvable

Rings

Rings

set $A$ together with two laws of composition called multiplication and addition which are written as product and sum respectively, satisfying following conditions, called ring

• $A$ is commutative group with respect to addition - unit element denoted by $0$
• $A$ is monoid with respect to multiplication - unit element denoted by $1$
• multiplication is distributive over addition, i.e. $$ \left( \forall x, y, z \in A \right) \left( (x+y)z = xz + yz \mbox{ \& } z(x+y) = zx + zy \right) $$
• do not assume $1\neq 0$

• can prove, e.g.,
  • $\left( \forall x \in A \right) \left( 0x = 0 \right)$ because $0x + x = 0x + 1x = (0+1)x = 1x = x$
  • if $1=0$, $A=\{0\}$ because $x = 1x = 0x = 0$
  • $\left( \forall x,y\in A \right) \left( (-x)y = -(xy) \right)$ because $xy + (-x)y = (x+ -x)y = 0y = 0$

subset of ring which itself is ring with same additive and multiplicative laws of composition, called subring

More on ring

subset $U$ of ring $A$ such that every element of $U$ has both left and right inverses, called group of units of $A$ or group of invertible elements of $A$, sometimes denoted by $A^\ast$

ring with $1\neq0$ and every nonzero element being invertible, called division ring

ring $A$ with $\left( \forall x,y \in A \right) \left( xy= yx \right)$, called commutative ring

subset $C\subset A$ of ring $A$ such that $$ C= \set{a\in A}{\forall x \in A, xa = ax} $$ is subring, and is called center of ring $A$

Fields

commutative division ring, called field

General distributivity

• general distributivity - for ring $A$, $\seq{x_i}_{i=1}^n\subset A$ and $\seq{y_i}_{i=1}^n\subset A$ $$ \left( \sum x_i \right) \left( \sum y_j \right) = \sum_i \sum_j x_iy_j $$

Ring examples

• for set $S$ and ring $A$, set of all mappings of $S$ into $A$ $\Map(S,A)$ whose addition and multiplication are defined as below, is ring $$ \begin{eqnarray*} & \left( \forall f,g\in \Map(S,A) \right) \left( \forall x\in S \right) \left( (f+g)(x) = f(x)+g(x) \right) & \\ & \left( \forall f,g\in \Map(S,A) \right) \left( \forall x\in S \right) \left( (fg)(x) = f(x)g(x) \right) & \end{eqnarray*} $$
  • additive and multiplicative unit elements of $\Map(S,A)$ are constant maps whose values are additive and multiplicative unit elements of $A$ respectively
  • $\Map(S,A)$ is commutative if and only if $A$ is commutative
  • for set $S$, $\Map(S,\reals)$ (page~) is a commutative ring
• for abelian group $M$, set $\End(M)$ of group homeomorphisms of $M$ into itself is ring with normal addition and mapping composition as multiplication
  • additive and multiplicative unit elements of $\End(M)$ are constant map whose value is the unit element of $M$ and identity mapping respectively
  • not commutative in general
• for ring $A$, set $A[X]$ of polynomials over $A$ is ring, ()
• for field $K$, $K^{n\times n}$, i.e., set of $n$-by-$n$ matrices with components in $K$, is ring
  • $\left(K^{n\times n}\right)^\ast$, i.e., multiplicative group of units of $K^{n\times n}$, consists of non-singular matrices, i.e., those whose determinants are nonzero

Group ring

for group $G$ and field $K$, set of all formal linear combinations $\sum_{x\in G} a_x x$ with $a_x\in K$ where $a_x$ are zero except finite number of them where addition is defined normally and multiplication is defined as $$ \left( \sum_{x\in G} a_x x \right) \left( \sum_{y\in G} b_y y \right) = \sum_{z\in G} \left( \sum_{xy=z} a_xb_y xy \right) $$ called group ring, denoted by $K[G]$

• $\sum_{xy=z} a_xb_y$ above defines what is called convolution product

Convolution product

for two functions $f,g$ on group $G$, convolution (product), denoted by $f\ast g$, defined by $$ (f\ast g)(z) = \sum_{xy=z} f(x)f(y) $$ as function on group $G$

• one may restrict this definition to functions which are $0$ except at finite number of elements

• for $f,g\in L^1(\reals)$, can define convolution product $f\ast g$ by $$ (f\ast g) (x) = \int_{\reals} f(x-y)g(y)dy $$
  • satisfies all axioms of ring except that there is not unit element
  • commutative (essentially because $\reals$ is commutative)
• more generally, for locally compact group $G$ wiht Haar measure $\mu$, can define convolution product by $$ (f\ast g) (x) = \int_{G} f(xy^{-1})g(y)d\mu(y) $$

Ideals of ring

subset $\ideal{a}$ of ring $A$ which is subgroup of additive group of $A$ with $A\ideal{a}\subset \ideal{a}$, called left ideal; indeed, $A\ideal{a} = \ideal{a}$ because $A$ has $1$; right ideal can be similarly defined, i.e., $\ideal{a} A = \ideal{a}$; subset which is both left and right ideal, called two-sided ideal or simply ideal

• for ring $A$, $(0)$ are $A$ itself area ideals

for ring $A$ and $a\in A$, left ideal $Aa$, called principal left ideal

• $a$, said to be generator of $\ideal{a}=Aa$ (over $A$)

$AaA$, called principal two-sided ideal where $$ AaA = \bigcup_{i=1}^\infty \bigsetl{\sum_{i=1}^n x_i a y_i}{x_i,y_i\in A} $$

only ideals of field are the field itself and zero ideal

Principle rings

commutative ring of which every ideal is principal and $1\neq0$, called principal ring

• $\integers$ (set of integers) is principal ring
• $k[X]$ (ring of polynomials) for field $k$ is principal ring
• ring of algebraic integers in number field $K$ is not necessarily principal
  • let $\ideal{p}$ be prime ideal, let $R_\ideal{p}$ be ring of all elements $a/b$ with $a,b\in R$ and $b\not\in\ideal{p}$, then $R_\ideal{p}$ is principal, with one prime ideal $\ideal{m}_\ideal{p}$ consisting of all elements $a/b$ as above but with $a\in\ideal{p}$
• let $A$ be set of entire functions on complex plane, then $A$ is commutative ring, and every finitely generated ideal is principal
  • given discrete set of complex numbers $\{z_i\}$ and nonnegative integers $\{m_i\}$, exists entire function $f$ having zeros at $z_i$ of multiplicity $m_i$ and no other zeros
  • every principal ideal is of form $Af$ for some such $f$
  • group of units $A^\ast$ in $A$ consists of functions having no zeros

Ideals as both additive and multiplicative monoids

• ideals form additive monoid
  • for left ideals $\ideal{a}$, $\ideal{b}$, $\ideal{c}$ of ring $A$, $\ideal{a}+\ideal{b}$ is left ideal, $(\ideal{a}+\ideal{b})+\ideal{c} =\ideal{a}+(\ideal{b}+\ideal{c})$, hence form additive monoid with $(0)$ as the unit elemenet
  • similarly for right ideals & two-sided ideals
• ideals form multiplicative monoid
  • for left ideals $\ideal{a}$, $\ideal{b}$, $\ideal{c}$ of ring $A$, define $\ideal{a}\ideal{b}$ as $$ \ideal{a}\ideal{b} = \bigcup_{i=1}^\infty \bigsetl{\sum_{i=1}^n x_i y_i}{x_i \in \ideal{a},y_i\in \ideal{b}} $$ then $\ideal{a}\ideal{b}$ is also left ideal, $(\ideal{a}\ideal{b})\ideal{c} =\ideal{a}(\ideal{b}\ideal{c})$, hence form multiplicative monoid with $A$ itself as the unit elemenet; for this reason, this unit element $A$, i.e., the ring itself, often written as $(1)$
  • similarly for right ideals & two-sided ideals
• ideal multiplication is also distributive over addition
• however, set of ideals does not form ring (because the additive monoid is not group)

Generators of ideal

for ring $A$ and $a_1,\ldots,a_n\subset A$, set of elements of $A$ of form $$ \sum_{i=1}^n x_i a_i $$ with $x_i \in A$, is left ideal, denoted by $(a_1,\ldots,a_n)$, called generators of the left ideal; similarly for right ideals

• above equal to smallest ideals containing $a_i$, i.e., intersection of all ideals containing $a_i$ $$ \cap_{a_1,\ldots, a_n\in\ideal{a}} \ideal{a} $$ - just like set ($\sigma$-)algebras in set theory

Entire rings

for ring $A$, $x,y\in A$ with $x\neq0$, $y\neq0$, and $xy=0$, said to be zero divisors

commutative ring with no zero divisors for which $1\neq0$, said to be entire; entire ring, sometimes called integral domain

every field is entire ring

Ring-homeomorphism

mapping of ring into ring $f:A\to B$ such that $f$ is monoid-homeomorphism for both additive and multiplicative structure on $A$ and $B$, i.e., $$ \left( \forall a, b \in A \right) \left( f(a+b) = f(a) + f(b) \mbox{ \& } f(ab) = f(a)f(b) \right) $$ and $$ f(1)=1 \mbox{ \& } f(0)=0 $$ called ring-homeomorphism; kernel, defined to be kernel of $f$ viewed as additive homeomorphism

• kernel of ring-homeomorphism $f:A\to B$ is ideal of $A$
• conversely, for ideal $\ideal{a}$, can construct factor ring $A/\ideal{a}$
• simply say “homeomorphism'' if reference to ring is clear

ring-homeomorphism from field into field is injective (due to )

Factor ring and canonical map

for ring $A$ and an ideal $\ideal{a} \subset A$, set of cosets $x+\ideal{a}$ for $x\in A$ combined with addition defined by viewing $A$ and $\ideal{a}$ as additive groups, multiplication defined by $(x+\ideal{a}) (y+\ideal{a}) = xy+\ideal{a},$ which satisfy all requirements for ring, called factor ring or residue class ring, denoted by $A/\ideal{a}$; cosets in $A/\ideal{a}$, called residue classes modulo \ideal{a}, and each coset $x+\ideal{a}$ called residue class of $x$ modulo \ideal{a}

• for ring $A$ and ideal $\ideal{a}$
  • for subset $S\subset \ideal{a}$, write $S \equiv 0 \Mod{\ideal{a}}$
  • for $x,y\in A$, if $x-y\in\ideal{a}$, write $x \equiv y \Mod{\ideal{a}}$
  • if $\ideal{a} = (a)$ for $a\in A$, for $x,y\in A$, if $x-y\in\ideal{a}$, write $x \equiv y \Mod{a}$

ring-homeomorphism of ring $A$ into factor ring $A/\ideal{a}$ $$ A \to A/\ideal{a} $$ called canonical map of $A$ into $A/\ideal{a}$

Factor ring induced ring-homeomorphism

for ring-homeomorphism $g:A\to A'$ whose kernel contains ideal $\ideal{a}$, exists unique ring-homeomorphism $g_\ast:A/\ideal{a} \to A'$ making diagram in the figure commutative, i.e., $g^\ast \circ f = g$ where $f$ is the ring canonical map $f:A\to A/\ideal{a}$

• the ring canonical map $f:A\to A/\ideal{a}$ is universal in category of homeomorphisms whose kernel contains $\ideal{a}$

Prime ideal and maximal ideal

for commutative ring $A$, ideal $\ideal{p}\neq A$ with $A/\ideal{p}$ entire, called prime ideal or just prime;

• equivalently, ideal $\ideal{p}\neq A$ is prime if and only if $\left( \forall x,y \in A \right) \left( xy \in \ideal{p} \Rightarrow x \in \ideal{p} \mbox{ or } y \in \ideal{p} \right)$

for commutative ring $A$, ideal $\ideal{m}\neq A$ such that $$ \left( \forall \mbox{ ideal } \ideal{a} \subset A \right) \left( \ideal{m} \subset \ideal{a} \Rightarrow \ideal{a} = A \right) $$ called maximal ideal

for commutative ring $A$

• every maximal ideal is prime
• every ideal is contained in some maximal ideal
• ideal $\{0\}$ is prime if and only if $A$ is entire
• ideal $\ideal{m}$ is maximal if and only if $A/\ideal{m}$ is field
• inverse image of prime ideal of commutative ring homeomorphism is prime

Embedding of ring

bijective ring-homeomorphism () is isomorphism

• indeed, for bijective ring-isomorphism $f:A\to B$, exists set-theoretic inverse $g:B\to A$ of $f$, which is ring-homeomorphism

image $f(A)$ of ring-homeomorphism $f:A\to B$ is subring of $B$

ring-isomorphism between $A$ and its image, established by injective ring-homeomorphism $f:A\to B$, called embedding of ring

for ring-homeomorphism $f:A\to A'$ and ideal $\ideal{a}'$ of $A'$, injective ring-homeomorphism $$ A/f^{-1}(\ideal{a}') \to A'/\ideal{a}' $$ called induced injective ring-homeomorphism

Characteristic of ring

• for ring $A$, consider ring-homeomorphism $$ \lambda:\integers \to A $$ such that $$ \lambda(n) = ne $$ where $e$ is multiplicative unit element of $A$
  • kernel of $\lambda$ is ideal $(n)$ for some $n\geq0$, i.e., ideal generated by some nonnegative integer $n$
  • hence, canonical injective ring-homeomorphism $\integers/n\integers \to A$, which is ring-isomorphism between $\integers/n\integers$ and subring of $A$
  • when $n\integers$ is prime ideal, exist two cases; either $n=0$ or $n=p$ for prime number $p$

ring $A$ with $\{0\}$ as prime ideal kernel above, said to have characteristic $0$; if prime ideal kernel is $p\integers$ for prime number $p$, $A$, said to have characteristic $p$, in which case, $A$ contains (isomorphic image of) $\integers/p\integers$ as subring, abbreviated by \primefield{p}

Prime fields and prime rings

• field $K$ has characteristic $0$ or $p$ for prime number $p$
• $K$ contains as subfield (isomorphic image of)
  • $\rationals$ if characteristic is $0$
  • $\primefield{p}$ if characteristic is $p$

in above cases, both $\rationals$ and $\primefield{p}$, called prime field (contained in $K$); since prime field is smallest subfield of $K$ containing $1$ having no automorphism other than identity, identify it with $\rationals$ or $\primefield{p}$ for each case

in above cases, prime ring (contained in $K$) means either integers $\integers$ if $K$ has characteristic $0$ or $\primefield{p}$ if $K$ has characteristic $p$

$\integers/n\integers$

• $\integers$ is ring
• every ideal of $\integers$ is principal, i.e., either $\{0\}$ or $n\integers$ for some $n\in\naturals$ (refer to page~)
• ideal of $\integers$ is prime if and only if is $p\integers$ for some prime number $p\in\naturals$
  • $p\integers$ is maximal ideal

$\integers/n\integers$, called ring of integers modulo $n$; abbreviated as $\mbox{mod }n$

• $\integers/p\integers$ for prime $p$ is field and denoted by \primefield{p}

Euler phi-function

for $n>1$, order of of $\integers/n\integers$, called Euler phi-function, denoted by $\varphi(n)$; if prime factorization of $n$ is $$ n = p_1^{e_1} \cdots p_r^{e_r} $$ with distinct $p_i$ and $e_i\geq1$ $$ \varphi(n) = p_1^{e_1-1} (p_1 - 1) \cdots p_r^{e_r-1} (p_r - 1) $$

for $x$ prime to $n$ $$ x^{\varphi(n)} \equiv 1 \Mod{n} $$

Chinese remainder theorem

for ring $A$ and $n$ ideals $\ideal{a}_1$, $\ideal{a}_n$ ($n\geq2$) with $\ideal{a}_i + \ideal{a}_j=A$ for all $i \neq j$ $$ \left( \forall x_1,\ldots, x_n \in A \right) \left( \exists x \in A \right) \left( \forall 1\leq i\leq n \right) \left( x \equiv x_i \Mod{\ideal{a}_i} \right) $$

for ring $A$, $n$ ideals $\ideal{a}_1$, $\ideal{a}_n$ ($n\geq2$) with $\ideal{a}_i + \ideal{a}_j=A$ for all $i \neq j$, and map of $A$ into product induced by canonical maps of $A$ onto $A/\ideal{a}_i$ for each factor, i.e., $$ f: A \to \prod A/\ideal{a}_i $$ $f$ is surjective and $\Ker f = \bigcap \ideal{a}_i$, hence, exists isomorphism $$ A/\cap \ideal{a}_i \isomorph \prod A/\ideal{a}_i $$

Isomorphism of endomorphisms of cyclic groups

for cyclic group $A$ of order $n$, endomorphisms of $A$ into $A$ with $x\mapsto kx$ for $k\in\integers$ induce

• ring isomorphism $$ \integers/n\integers \isomorph \End(A) $$
• group isomorphism $$ (\integers/n\integers)^\ast \isomorph \Aut(A) $$

where $(\integers/n\integers)^\ast$ denotes group of units of $\integers/n\integers$ ()

• e.g., for group of $n$-th roots of unity in $\complexes$, all automorphisms are given by $$ \xi \mapsto \xi^k $$ for $k\in(\integers/n\integers)^\ast$

Irreducibility and factorial rings

for entire ring $A$, non-unit non-zero element $a\in A$ with $$ \left( \forall b, c\in A \right) \left( a = bc \Rightarrow b \mbox{ or } c \mbox{ is unit} \right) $$ said to be irreducible

for entire ring $A$, element $a\in A$ for which, exists unit $u$ and irreducible elements, $p_1$, , $p_r$ in $A$ such that $$ a = u \prod p_i $$ and this expression is unique up to permutation and multiplications by units, said to have unique factorization into irreducible elements

entire ring with every non-zero element has unique factorial into irreducible elements, called factorial ring or unique factorization ring

Greatest common divisor

for entire ring $A$ and nonzero elements $a,b\in A$, $a$ said to divide $b$ if exists $c\in A$ such that $ac=b$, denoted by $a|b$

for entire ring $A$ and $a,b\in A$, $d\in A$ which divides $a$ and $b$ and satisfies $$ \left( \forall c \in A \right) \left( c|a \mbox{ \& } c|b \Rightarrow c | d \right) $$ called greatest common divisor (g.c.d.) of $a$ and $b$

for principal entire ring $A$ and nonzero $a,b\in A$, $c\in A$ with $(a,b) = (c)$ is g.c.d. of $a$ and $b$

principal entire ring is factorial

Polynomials

Why (ring of) polynomials?

• lays ground work for polynomials in general
• needs polynomials over arbitrary rings for diverse purposes
  • polynomials over finite field which cannot be identified with polynomial functions in that field
  • polynomials with integer coefficients; reduce them mod $p$ for prime $p$
  • polynomials over arbitrary commutative rings
  • rings of polynomial differential operators for algebraic geometry & analysis
• e.g., ring learning with errors (RLWE) for cryptographic algorithms

Ring of polynomials

• exist many ways to define polynomials over commutative ring; here's one

for ring $A$, set of functions from monoid $S = \set{X^r}{r\in\integers, r\geq0}$ into $A$ which are equal to $0$ except finite number of elements of $S$, called polynomials over $A$, denoted by $A[X]$

• for every $a\in A$, define function which has value $a$ on $X^n$, and value $0$ for every other element of $S$, by $aX^r$
• then, a polynomial can be uniquely written as $$ f(X) = a_0X^0 + \cdots + a_nX^n $$ for some $n\in\integers_+$, $a_i\in A$
• $a_i$, called coefficients of $f$

Polynomial functions

for two rings $A$ and $B$ with $A\subset B$ and $f\in A[X]$ with $f(X) = a_0 + a_1 X + \cdots + a_nX^n$, map $f_B: B\to B$ defined by $$ f_B(x) = a_0 + a_1 x + \cdots + a_n x^n $$ called polynomial function associated with $f(X)$

for two rings $A$ and $B$ with $A\subset B$ and $b\in B$, ring homeomorphism from $A[X]$ into $B$ with association, $\ev_b:f\mapsto f(b)$, called evaluation homeomorphism, said to be obtained by substituting $b$ for $X$ in $f$

• hence, for $x\in B$, subring $A[x]$ of $B$ generated by $x$ over $A$ is ring of all polynomial values $f(x)$ for $f\in A[X]$

for two rings $A$ and $B$ with $A\subset B$, if $x\in B$ makes evaluation homeomorphism $\ev_x:f\mapsto f(x)$ isomorphic, $x$, said to be transcendental over $A$ or variable over $A$

• in particular, $X$ is variable over $A$

Polynomial examples

• consider $\alpha=\sqrt{2}$ and $\bigset{a+b\alpha}{a,b\in\integers}$, subring of $\integers[\alpha]\subset \reals$ generated by $\alpha$.
  • $\alpha$ is not transcendental because $f(\alpha)=0$ for $f(X)=X^2-1$
  • hence kernel of evaluation map of $\integers[X]$ into $\integers[\alpha]$ is not injective, hence not isomorphism
  • indeed $$ \integers[\alpha] = \bigset{a+b\alpha}{a,b\in\integers} $$
• consider $\primefield{p}$ for prime number $p$
  • $f(X) = X^p - X\in \primefield{p}[X]$ is not zero polynomial, but because $x^{p-1} \equiv 1$ for every nonzero $x\in\primefield{p}$ by (Euler's theorem), $x^p\equiv x$ for every $x\in\primefield{p}$, thus for polynomial function, $f_{\primefield{p}}$, $f_{\primefield{p}}(x)=0$ for every $x$ in $\primefield{p}$
  • i.e., non-zero polynomial induces zero polynomial function

Reduction map

• for homeomorphism $\varphi:A\to B$ of commutative rings, exists associated homeomorphisms of polynomial rings $A[X]\to B[X]$ such that $$ f(X) = \sum a_i X^i \mapsto \sum \varphi(a_i) X^i = (\varphi f)(X) $$

above ring homeomorphism $f\mapsto \varphi f$, called reduction map

• e.g., for complex conjugate $\varphi: \complexes \to \complexes$, homeomorphism of $\complexes[X]$ into itself can be obtained by reduction map $f \mapsto \varphi f$, which is complex conjugate of polynomials with complex coefficients

for prime ideal $\ideal{p}$ of ring $A$ and surjective canonical map $\varphi: A \to A/\ideal{p}$, reduction map $\varphi f$ for $f\in A[X]$, sometimes called reduction of $f$ modulo \ideal{p}

Basic properties of polynomials in one variable

for set of all polynomials in one variable of nonnegative degrees $A[X]$ with commutative ring $A$ $$ \begin{eqnarray*} &=& \left( \forall f,g \in A[X] \mbox{ with leading coefficients of } g \mbox{ unit in }A \right) \\ && \left( \exists q, r \in A[X] \mbox{ with } \deg r < \deg g \right) \left( f = qg + r \right) \end{eqnarray*} $$

polynomial ring in one variable $k[X]$ with field $k$ is principal

polynomial ring in one variable $k[X]$ with field $k$ is factorial

Constant, monic, and irreducible polynomials

$k \in k[X]$ with field $k$, called constant polynomial; $f(x) \in k[X]$ with leading coefficient $1$, called monic polynomial

polynomial $f(x)\in k[X]$ such that $$ \left( \forall g(X), h(X) \in k[X] \right) \left( f(X) = g(X)h(X) \Rightarrow g(X) \in k \mbox{ or } h(X) \in k \right) $$ said to be irreducible

Roots or zeros of polynomials

for commutative ring $B$, its subring $A\subset B$, and $f(x)\in A[X]$ in one variable, $b\in B$ satisfying $$ f(b) = 0 $$ called root or zero of $f$

for field $k$, polynomial $f\in k[X]$ in one variable of degree $n\geq 0$ has at most $n$ roots in $k$; if $a$ is root of $f$ in $k$, $X-a$ divides $f(X)$

Induction of zero functions

for field $k$ and infinite subset $T\subset k$, if polynomial $f\in k[X]$ in one variable over $k$ satisfies $$ \left( \forall a \in k \right) \left( f(a) =0 \right) $$ then $f(0)=0$, i.e., $f$ induces zero function

for field $k$ and $n$ infinite subsets of $k$, $\seq{S_i}_{i=1}^n$, if polynomial in $n$ variables over field $k$ satisfies $$ \left( \forall a_i\in S_i \mbox{ for } 1\leq i \leq n \right) \left( f(a_1,\ldots,a_n)=0 \right) $$ then $f=0$, i.e., $f$ induces zero function

if polynomial in $n$ variables over infinite field $k$ induces zero function in $k^{(n)}$, $f=0$

if polynomial in $n$ variables over finite field $k$ of order $q$, degree of which in each variable is less than $q$, induces zero function in $k^{(n)}$, $f=0$

Reduced polynomials and uniqueness

• for field $k$ with $q$ elements, polynomial in $n$ variables over $k$ can be expressed as $$ f(X_1,\ldots,X_n) = \sum a_i X_1^{\nu_{i,1}} \cdots X_n^{\nu_{i,n}} $$ for finite sequence, $\seqscr{a_i}{i=1}{m}$, and $\seqscr{\nu_{i,1}}{i=1}{m}$, , $\seqscr{\nu_{i,n}}{i=1}{m}$ where $a_i\in k$ and $\nu_{i,j} \geq 0$
• because $X_i^q=X_i$ for any $X_i$, any $\nu_{i,j}\geq q$ can be (repeatedly) replaced by $\nu_{i,j}-(q-1)$, hence $f$ can be rewritten as $$ f(X_1,\ldots,X_n) = \sum a_i X_1^{\mu_{i,1}} \cdots X_n^{\mu_{i,n}} $$ where $0\leq \mu_{i,j} < q$ for all $i,j$

above polynomial, called reduced polynomial, denoted by $f^\ast$

for field $k$ with $q$ elements, reduced polynomial is unique (by )

Multiplicative subgroups and $n$-th roots of unity

for field $k$, subgroup of group $k^\ast=k\sim \{0\}$, called multiplicative subgroup of $k$

finite multiplicative subgroup of field is cyclic

multiplicative subgroup of finite field is cyclic

generator for group of $n$-th roots of unity, called primitive $n$-th root of unity; group of roots of unity, denoted by $\mu$; group of roots of unity in field $k$, denoted by $\mu(k)$

Algebraic closedness

field $k$, for which every polynomial in $k[X]$ of positive degree has root in $k$, said to be algebraically closed

• e.g., complex numbers are algebraically closed
• every field is contained in some algebraically closed field ()
• for algebraically closed field $k$
  • (of course) every irreducible polynomial in $k[X]$ is of degree $1$
  • unique factorization of polynomial of nonnegative degree can be written in form $$ f(X) = c \prod_{i=1}^{r} (X-\alpha_i)^{m_i} $$ with nonzero $c\in k$, distinct roots, $\alpha_1,\ldots,\alpha_r \in k$, and $m_1,\ldots,m_r \in \naturals$

Derivatives of polynomials

for polynomial $f(X) = a_nX^n + \cdots + a_1 X + a_0 \in A[X]$ with commutative ring $A$, map $D:A[X] \to A[X]$ defined by $$ Df(X) = na_n X^{n-1} + \cdots + a_1 $$ called derivative of polynomial, denoted by $f'(X)$;

• for $f,g\in A[X]$ with commutative ring $A$, and $a\in A$ $$ (f+g)' = f' + g' \quad \mbox{\&} \quad (fg)' = f'g + fg' \quad \mbox{\&} \quad (af)' = af' $$

Multiple roots and multiplicity

• nonzero polynomial $f(X)\in k[X]$ in one variable over field $k$ having $a\in k$ as root can be written of form $$ f(X) = (X-a)^m g(X) $$ with some polynomial $g(X)\in A[X]$ relatively prime to $(X-a)$ (hence, $g(a)\neq0$)

above, $m$, called multiplicity of $a$ in $f$; $a$, said to be multiple root of $f$ if $m>1$

for polynomial $f$ of one variable over field $k$, $a\in k$ is multiple root of $f$ if and only if $f(a)=0$ and $f'(a)=0$

for polynomial $f\in K[X]$ over field $K$ of positive degree, $f'\neq0$ if $K$ has characteristic $0$; if $K$ has characteristic $p>0$, $f'=0$ if and only if $$ f(X) = \sum_{\nu=1}^n a_\nu X^\nu $$ where $p$ divides each integer $\nu$ whenever $a_\nu\neq0$

Frobenius endomorphism

• homeomorphism of $K$ into itself $x\mapsto x^p$ has trivial kernel, hence injective
• hence, iterating $r\geq 1$ times yields endomorphism, $x\mapsto x^{p^r}$

for field $K$, prime number $p$, and $r\geq1$, endomorphism of $K$ into itself, $x\mapsto x^{p^r}$, called Frobenius endomorphism

Roots with multiplicity $p^r$ in fields having characteristic $p$

• for field $K$ having characteristic $p$
  • $p | {p \choose \nu}$ for all $0< \nu < p$ because $p$ is prime, hence, for every $a,b\in K$ $$ (a+b)^p = a^p + b^p $$
  • applying this resurvely $r$ times yields $$ (a+b)^{p^r} = (a^p + b^p)^{p^{r-1}} = (a^{p^2} + b^{p^2})^{p^{r-2}} = \cdots = a^{p^r} + b^{p^r} $$ hence $$ (X-a)^{p^r} = X^{p^r} - a^{p^r} $$
  • if $a,c\in K$ satisfy $a^{p^r} = c$ $$ X^{p^r} - c = X^{p^r} - a^{p^r} = (X-a)^{p^r} $$ hence, polynomial $X^{p^r}-c$ has precisely one root $a$ of multiplicity $p^r$!

Algebraic Extension

Algebraic extension

• will show
  • for polynomial over field, always exists some extension of that field where the polynomial has root
  • existence of algebraic closure for every field

Extension of field

for field $E$ and its subfield $F\subset E$, $E$ said to be extension field of $F$, (sometimes) denoted by $E/F$ (which should not confused with factor group)

• can view $E$ as vector space over $F$
• if dimension of the vector space is finite, extension called finite extension of $F$
• if infinite, called infinite extension of $F$

Algebraic over field

for field $E$ and its subfield $F\subset E$, $\alpha\in E$ satisfying $$ \left( \exists a_0,\ldots, a_n \mbox{ with not all } a_i \mbox{ zero} \right) \left( a_0 + a_1\alpha + \cdots + a_n \alpha^n=0 \right) $$ said to be algebraic over $F$

• for algebraic $\alpha\neq0$, can always find such equation like above that $a_0\neq0$

• equivalent statements to
  • exists homeomorphism $\varphi: F[X] \to E$ such that $$ \left(\forall x\in F\right) \left(\varphi(x) = x\right) \mbox{ \& } \varphi(X) = \alpha \mbox{ \& } \Ker \varphi \neq \{0\} $$
  • exists evaluation homeomorphism $\ev_\alpha: F[X] \to E$ with nonzero kernel (refer to for definition of evaluation homeomorphism)
• in which case, $\Ker \varphi$ is principal ideal (by ), hence generated by single element, thus exists nonzero $p(X) \in F[X]$ (with normalized leading coefficient being $1$) so that $$ F[X] / (p(X)) \isomorph F[\alpha] $$
• $F[\alpha]$ entire (), hence $p(X)$ irreducible (refer to )

normalized $p(X)$ (i.e., with leading coefficient being $1$) uniquely determined by $\alpha$, called THE irreducible polynomial of $\alpha$ over $F$, denoted by $\Irr(\alpha, F, X)$

Algebraic extensions

for field $F$, its extension field every element of which is algebraic over $F$, said to be algebraic extension of $F$

for field $F$, every finite extension field of $F$ is algebraic over $F$

• converse is not true, e.g., subfield of complex numbers consisting of algebraic numbers over $\rationals$ is infinite extension of $\rationals$

Dimension of extensions

for field $F$ and its extension field $E$, dimension of $E$ as vector space over $F$, called dimension of $E$ over $F$, denoted by \dimext{E}{F}

for field $k$ and its extension fields $F$ and $E$ with $k\subset F\subset E$ $\dimext{E}{k} = \dimext{E}{F} \dimext{F}{k}$

• if $\seqscr{x_i}{i\in I}{}$ is basis for $F$ over $k$, and $\seqscr{y_j}{j\in J}{}$ is basis for $E$ over $F$, $\seqscr{x_iy_j}{(i,j)\in I\times J}{}$ is basis for $E$ over $k$

for field $k$ and its extension fields $F$ & $E$ with $k\subset F\subset E$, $E/k$ is finite if and only if both $F/k$ and $E/F$ are finite

Generation of field extensions

for field $k$, its extension field $E$, and $\alpha_1,\ldots, \alpha_n \in E$, smallest subfield containing $k$ and $\alpha_1$, , $\alpha_n$, said to be finitely generated over $k$ by $\alpha_1$, \ldots, $\alpha_n$, denoted by $k(\alpha_1,\ldots, \alpha_n)$

• $k(\alpha_1,\ldots, \alpha_n)$ consists of all quotients $f(\alpha_1,\ldots,\alpha_n)/g(\alpha_1,\ldots, \alpha_n)$ where $f,g\in k[X]$ and $g(\alpha_1,\ldots, \alpha_n)\neq0$, i.e. $$ \begin{eqnarray*} &=& k(\alpha_1,\ldots,\alpha_n) \\ &=& \bigset{f(\alpha_1,\ldots, \alpha_n)/g(\alpha_1,\ldots,\alpha_n)}{f,g\in f[X], g(\alpha_1,\ldots,\alpha_n)\neq0} \end{eqnarray*} $$
• any field extension $E$ over $k$ is union of smallest subfields containing $\alpha_1,\ldots, \alpha_n$ where $\alpha_1,\ldots, \alpha_n$ range over finite set of elements of $E$, i.e. $$ E = \bigcup_{n\in\naturals} \bigcup_{\alpha_1, \ldots, \alpha_n \in E} k(\alpha_1,\ldots,\alpha_n) $$

every finite extension of field is finitely generated

Tower of fields

sequence of extension fields $$ F_1 \subset F_2 \subset \cdots \subset F_n $$ called tower of fields

tower of fields, said to be finite if and only if each step of extensions is finite

Algebraicness of finitely generated subfields

for field $k$, its extension field $E$, and $\alpha\in E$ being algebraic over $k$ $$ k(\alpha) = k[\alpha] $$ and $$ [k(\alpha):k] = \deg \Irr(\alpha, k, X) $$ hence $k(\alpha)$ is finite extension of $k$, thus algebraic extension over $k$ (by )

for field $k$, its extension field $F$, and $\alpha\in E$ being algebraic over $k$ where $k(\alpha)$ and $F$ are subfields of common field, $\alpha$ is algebraic over $F$

• indeed, $\Irr(\alpha,k,X)$ has a fortiori coefficients in $F$

• assume tower of fields $$ k \subset k(\alpha_1) \subset k(\alpha_1, \alpha_2) \subset \cdots \subset k(\alpha_1,\ldots, \alpha_n) $$ where $\alpha_i$ is algebraic over $k$
• then, $\alpha_{i+1}$ is algebraic over $k(\alpha_1,\ldots,\alpha_i)$ (by )

for field $k$ and $\alpha_1$, , $\alpha_n$ being algebraic over $k$, $E=k(\alpha_1,\ldots,\alpha_n)$ is finitely algebraic over $k$ (due to , , and ). Indeed, $E = k[\alpha_1, \ldots, \alpha_n]$ and $$ \begin{eqnarray*} [k(\alpha_1,\ldots,\alpha_n):k] &=& \deg \Irr(\alpha_1,k,X) \deg \Irr(\alpha_2,k(\alpha_1),X) \\ && \cdots \deg \Irr(\alpha_n, k(\alpha_1,\ldots,\alpha_{n-1}), X), \end{eqnarray*} $$

Compositum of subfields and lifting

for field $k$ and its extension fields $E$ and $F$, which are subfields of common field $L$, smallest subfield of $L$ containing both $E$ and $F$, called compositum of $E$ and $F$ in $L$, denoted by $EF$

• cannot define compositum if $E$ and $F$ are not embedded in common field $L$

• could define compositum of set of subfields of $L$ as smallest subfield containing subfields in the set

extension $E$ of $k$ is compositum of all its finitely generated subfields over $k$, i.e., $E = \bigcup_{n\in\naturals} \bigcup_{\alpha_1, \ldots, \alpha_n \in E} k(\alpha_1,\ldots,\alpha_n)$

Lifting

extension $EF$ of $F$, called translation of $E$ to $F$ or lifting of $E$ to $F$

• often draw diagram as in the figure

Finite generation of compositum

for field $k$, its extension field $F$, and $E = k(\alpha_1,\ldots,\alpha_n)$ where both $E$ and $F$ are contained in common field $L$, $$ EF = F(\alpha_1, \ldots, \alpha_n) $$ i.e., compositum $EF$ is finitely generated over $F$

• refer to diagra in the figure

Distinguished classes

for field $k$, class $\classk{C}$ of extension fields satisfying

• for tower of fields $k\subset F\subset E$, extension $k\subset E$ is in $\classk{C}$ if and only if both $k\subset F$ and $F\subset E$ are in $\classk{C}$
• if $k\subset E$ is in $\classk{C}$, $F$ is any extension of $k$, and both $E$ and $F$ are subfields of common field, then $F\subset EF$ is in $\classk{C}$

said to be distinguished; the figure illustrates these two properties, which imply the following property

• if $k\subset F$ and $k\subset E$ are in $\classk{C}$ and both $E$ and $F$ are subfields of common field, $k\subset EF$ is in $\classk{C}$

Both algebraic and finite extensions are distinguished

class of algebraic extensions is distinguished, so is class of finite extensions

• true that finitely generated extensions form distinguished class (not necessarily algebraic extensions or finite extensions)

Field embedding and embedding extension

for two fields $F$ and $L$, injective homeomorphism $\sigma:F\to L$, called embedding of $F$ into $L$; then (of course) $\sigma$ induces isomorphism of $F$ with its image $\sigma F$

for field embedding $\sigma:F\to L$, field extension $F\subset E$, and embedding $\tau:E\to L$ whose restriction to $F$ being equal to $\sigma$, said to be over $\sigma$ or extend $\sigma$; if $\sigma$ is identity, embedding $\tau$, called embedding of $E$ over $F$; diagrams in the figure show these embedding extensions

• assuming $F$, $E$, $\sigma$, and $\tau$ same as in , if $\alpha\in E$ is root of $f\in F[X]$, then $\alpha^\tau$ is root of $f^\sigma$ for if $f(X) = \sum_{i=0}^n a_i X^i$, then $f(\alpha) = \sum_{i=0}^n a_i \alpha^i = 0$, and $0 = f(\alpha)^\tau = \sum_{i=0}^n (a_i^\tau ) (\alpha^\tau)^i = \sum_{i=0}^n a_i^\sigma (\alpha^\tau)^i = f^\sigma(\alpha^\tau)$

Embedding of field extensions

for field $k$ and its algebraic extension $E$, embedding of $E$ into itself over $k$ is isomorphism

for field $k$ and its field extensions $E$ and $F$ contained in common field, $$ E[F] = F[E] = \bigcup_{n=1}^\infty \bigset{e_1f_1 + \cdots + e_nf_n}{e_i\in E, f_i\in F} $$ and $EF$ is field of quotients of these elements

for field $k$, its field extensions $E_1$ and $E_2$ contained in commen field $E$, and embedding $\sigma:E\to L$ for field $L$, $$ \sigma(E_1 E_2) = \sigma(E_1) \sigma(E_2) $$

Existence of roots of irreducible polynomial

• assume $p(X) \in k[X]$ irreducible polynomial and consider canonical map, which is ring homeomorphism $$ \sigma: k[X] \to k[X] / ((p(X)) $$
• consider $\Ker \restrict{\sigma}{k}$
  • every kernel of ring homeomorphism is ideal, hence if nonzero $a \in \Ker \restrict{\sigma}{k}$, $1\in \Ker \restrict{\sigma}{k}$ because $a^{-1} \in \Ker \restrict{\sigma}{k}$, but $1\not\in (p(X))$
  • thus, $\Ker \restrict{\sigma}{k} = \{0\}$, hence $p^\sigma\neq0$
• now for $\alpha = X^\sigma$ $$ p^\sigma(\alpha) = p^\sigma(X^\sigma) = (p(X))^\sigma = 0 $$
• thus, $\alpha$ is algebraic in $k^\sigma$, i.e., $\alpha \in k[X]^\sigma$ is root of $p^\sigma$ in $k^\sigma(\alpha)$

for field $k$ and irreducible $p(X)\in k[X]$ with $\deg p \geq 1$, exist field $L$ and homeomorphism $\sigma:k \to L$ such that $p^\sigma$ with $\deg p^\sigma \geq 1$ has root in field extension of $k^\sigma$

Existence of algebraically closed algebraic field extensions

for field $k$ and $f\in k[X]$ with $\deg f \geq 1$, exists extension of $k$ in which $f$ has root

for field $k$ and $f_1$, , $f_n$ $\in$ $k[X]$ with $\deg f_i \geq 1$, exists extension of $k$ in which every $f_i$ has root

for every field $k$, exists algebraically closed extension of $k$

for every field $k$, exists algebraically closed algebraic extension of $k$

Isomorphism between algebraically closed algebraic extensions

for field, $k$, $\alpha$ being algebraic over $k$, algebraically closed field, $L$, and embedding, $\sigma:k\to L$, # possible embedding extensions of $\sigma$ to $k(\alpha)$ in $L$ is equal to # distinct roots of $\Irr(\alpha,k,X)$, hence no greater than # roots of $\Irr(\alpha,k,X)$

for field, $k$, its algebraic extensions, $E$, algebraically closed field, $L$, and embedding, $\sigma:k\to L$, exists embedding extension of $\sigma$ to $E$ in $L$; if $E$ is algebraically closed and $L$ is algebraic over $k^\sigma$, every such embedding extension is isomorphism of $E$ onto $L$

for field, $k$, and its algebraically closed algebraic extensions, $E$ and $E'$, exists isomorphism bewteen $E$ and $E'$ which induces identity on $k$, i.e. $$ \tau: E \to E' $$ where $\restrict{\tau}{k}$ is identity

• thus, algebraically closed algebraic extension is determined up to isomorphism

Algebraic closure

for field, $k$, algebraically closed algebraic extension of $k$, which is determined up to isomorphism, called algebraic closure of $k$, frequently denoted by \algclosure{k}

• examples
  • complex conjugation is automorphism of $\complexes$ (which is the only continuous automorphism of $\complexes$)
  • subfield of $\complexes$ consisting of all numbers which are algebraic over $\rationals$ is algebraic closure of $\rationals$, i.e., $\algclosure{\rationals}$
  • $\algclosure{\rationals} \neq \complexes$
  • $\algclosure{\reals} = \complexes$
  • \algclosure{\rationals}\ is countable

algebraic closure of finite field is countable

for infinite field, $k$, every algebraic extension of $k$ has same cardinality as $k$

Splitting fields

for field, $k$, and $f\in k[X]$ with $\deg f \geq 1$, field extension, $K$, of $k$, $f$ splits into linear factors in which, i.e., $$ f(X) = c (X-\alpha_1) \cdots (X-\alpha_n) $$ and which is finitely generated over $k$ by $\alpha_1$, , $\alpha_n$ (hence $K=k(\alpha_1, \ldots, \alpha_n)$), called splitting field of $f$

• for field, $k$, every $f\in k[X]$ has splitting field in $\algclosure{k}$

for field, $k$, $f\in k[X]$ with $\deg f \geq1$, and two splitting fields of $f$, $K$ and $E$, exists isomorphism between $K$ and $E$; if $k\subset K\subset \algclosure{k}$, every embedding of $E$ into $\algclosure{k}$ over $k$ is isomorphism of $E$ onto $K$

Splitting fields for family of polynomials

for field, $k$, index set, $\Lambda$, and indexed family of polynomials, $\set{f_\lambda\in k[X]}{\lambda \in \Lambda, \deg f_\lambda \geq1}$, extension field of $k$, every $f_\lambda$ splits into linear factors in which and which is generated by all roots of all polynomials, $f_\lambda$, called splitting field for family of polynomials

• in most applications, deal with finite $\Lambda$
• becoming increasingly important to consider infinite algebraic extensions
• various proofs would not be simpler if restricted ourselves to finite cases

for field, $k$, index set, $\Lambda$, and two splitting fields, $K$ and $E$, for family of polynomials, $\set{f_\lambda\in k[X]}{\lambda \in \Lambda, \deg f_\lambda \geq1}$, every embedding of $E$ into $\algclosure{K}$ over $k$ is isomorphism of $E$ onto $K$

Normal extensions

for field, $k$, and its algebraic extension, $K$, with $k\subset K\subset \algclosure{k}$, following statements are equivalent

• every embedding of $K$ into $\algclosure{k}$ over $k$ induces automorphism
• $K$ is splitting field of family of polynomials in $k[X]$
• every irreducible polynomial of $k[X]$ which has root in $K$ splits into linear factors in $K$

for field, $k$, and its algebraic extension, $K$, with $k\subset K\subset \algclosure{k}$, satisfying properties in , said to be normal

• not true that class of normal extensions is distinguished
  • e.g., below tower of fields is tower of normal extensions $$ \rationals \subset \rationals(\sqrt{2}) \subset \rationals(\sqrt[4]{2}) $$
  • but, extension $\rationals \subset \rationals(\sqrt[4]{2})$ is not normal because complex roots of $X^4-2$ are not in $\rationals(\sqrt[4]{2})$

Retention of normality of extensions

normal extensions remain normal under lifting; if $k\subset E\subset K$ and $K$ is normal over $k$, $K$ is normal over $E$; if $K_1$ and $K_2$ are normal over $k$ and are contained in common field, $K_1K_2$ is normal over $k$, and so is $K_1\cap K_2$

Separable degree of field extensions

• for field, $F$, and its algebraic extension, $E$
  • let $L$ be algebraically closed field and assume embedding, $\sigma:F\to L$
    • exists embedding extension of $\sigma$ to $E$ in $L$ by
    • such $\sigma$ maps $E$ on subfield of $L$ which is algebraic over $F^\sigma$
    • hence, $E^\sigma$ is contained in algebraic closure of $F^\sigma$ which is contained in $L$
    • will assume that $L$ is the algebraic closure of $F^\sigma$
  • let $L'$ be another algebraically closed field and assume another embedding, $\tau:F\to L'$ - assume as before that $L'$ is algebraic closure of $F^\tau$
  • then implies, exists isomorphism, $\lambda:L\to L'$ extending $\tau\circ \sigma^{-1}$ applied to $F^\sigma$
  • let $S_\sigma$ & $S_\tau$ be sets of embedding extensions of $\sigma$ to $E$ in $L$ and $L'$ respectively
  • then $\lambda$ induces map from $S_\sigma$ into $S_\tau$ with $\tilde{\sigma} \mapsto \lambda \circ \tilde{\sigma}$ and $\lambda^{-1}$ induces inverse map from $S_\tau$ into $S_\sigma$, hence exists bijection between $S_\sigma$ and $S_\tau$, hence have same cardinality

above cardinality only depends on extension $E/F$, called separable degree of $E$ over $F$, denoted by $[E:F]_s$

Multiplicativity of and upper bound on separable degree of field extensions

for tower of algebraic field extensions, $k\subset F\subset E$, $$ [E:k]_s = [E:F]_s [F:k]_s $$

for finite algebraic field extension, $k\subset E$ $$ [E:k]_s \leq [E:k] $$

• i.e., separable degree is at most equal to degree (i.e., dimension) of field extension

for tower of algebraic field extensions, $k\subset F\subset E$, with $[E:k]<\infty$ $$ [E:k]_s = [E:k] $$ holds if and only if corresponding equality holds in every step of tower, i.e., for $E/F$ and $F/k$

Finite separable field extensions

for finite algebraic field extension, $E/k$, with $[E:k]_s=[E:k]$, $E$, said to be separable over $k$

for field, $k$, $\alpha$, which is algebraic over $k$ with $k(\alpha)$ being separable over $k$, said to be separable over $k$

for field, $k$, $\alpha$, which is algebraic over $k$, is separable over $k$ if and only if $\Irr(\alpha,k,X)$ has no multiple roots

for field, $k$, $f\in k[X]$ with no multiple roots, said to be separable

for tower of algebraic field extensions, $k\subset F\subset K$, if $\alpha \in K$ is separable over $k$, then $\alpha$ is separable over $F$

for finite field extension, $E/k$, $E$ is separable over $k$ if and only if every element of $E$ is separable over $k$

Arbitrary separable field extensions

for (not necessarily finite) field extension, $E/k$, $E$, of which every finitely generated subextension is separable over $k$, i.e., $$ \left( \forall n\in\naturals\ \& \ \alpha_1,\ldots, \alpha_n \in E \right) \left( k(\alpha_1, \ldots, \alpha_n) \mbox{ is separable over } k \right) $$ said to be separable over $k$

for algebraic extension, $E/k$, $E$, which is generated by family of elements, $\{\alpha_\lambda\}_{\lambda\in\Lambda}$, with every $\alpha_\lambda$ is separable over $k$, is separable over $k$

separable extensions form distinguished class of extensions

Separable closure and conjugates

for field, $k$, compositum of all separable extensions of $k$ in given algebraic closure $\algclosure{k}$, called separable closure of $k$, denoted by $k^\mathrm{s}$ or \sepclosure{k}

for algebraic field extension, $E/k$, and embedding of $E$, $\sigma$, in $\algclosure{k}$ over $k$, $E^\sigma$, called conjugate of $E$ in \algclosure{k}

• smallest normal extension of $k$ containing $E$ is compositum of all conjugates of $E$ in $\algclosure{E}$

for field, $k$, $\alpha$ being algebraic over $k$, and distinct embeddings, $\sigma_1$, , $\sigma_r$ of $k(\alpha)$ into $\algclosure{k}$ over $k$, $\alpha^{\sigma_1}$, , $\alpha^{\sigma_r}$, called conjugates of $\alpha$ in \algclosure{k}

• $\alpha^{\sigma_1}$, , $\alpha^{\sigma_r}$ are simply distinct roots of $\Irr(\alpha, k, X)$
• smallest normal extension of $k$ containing one of these conjugates is simply $k(\alpha^{\sigma_1}, \ldots, \alpha^{\sigma_r})$

Prime element theorem

for finite algebraic field extension, $E/k$, exists $\alpha\in E$ such that $E=k(\alpha)$ if and only if exists only finite # fields, $F$, such that $k\subset F\subset E$; if $E$ is separable over $k$, exists such element, $\alpha$

for finite algebraic field extension, $E/k$, $\alpha\in E$ with $E=k(\alpha)$, called primitive element of $E$ over $k$

Finite fields

for every prime number, $p$, and integer, $n\geq1$, exists finite field of order $p^n$, denoted by \finitefield{p}{n}, uniquely determined as subfield of algebraic closure, $\algclosure{\primefield{p}}$, which is splitting field of polynomial $$ f_{p,n}(X) = X^{p^n} - X $$ and whose elements are roots of $f_{p,n}$

for every finite field, $F$, exist prime number, $p$, and integer, $n\geq1$, such that $F=\finitefield{p}{n}$

for finite field, , and integer, $m\geq1$, exists one and only one extension of degree, $m$, which is

multiplicative group of finite field is cyclic

Automorphisms of finite fields

mapping $$ \frobmap{p}{n}: \finitefield{p}{n} \to \finitefield{p}{n} $$ defined by $x\mapsto x^p$, called Frobenius mapping

• is (ring) homeomorphism with $\Ker \frobmap{p}{n} = \{0\}$ since is field, thus is injective (), and surjective because is finite,
• thus, is isomorphism leaving \primefield{p}\ fixed

group of automorphisms of is cyclic of degree $n$, generated by

for prime number, $p$, and integers, $m,n\geq1$, in any $\algclosure{\primefield{p}}$, is contained in if and only if $n$ divides $m$, i.e., exists $d\in\integers$ such that $m=dn$, in which case, is normal and separable over group of automorphisms of over is cyclic of order, $d$, generated by $\frobmap{p}{m}^n$

Galois Theory

What we will do to appreciate Galois theory

• study
  • group of automorphisms of finite (and infinite) Galois extension (at length)
  • give examples, e.g., cyclotomic extensions, abelian extensions, (even) non-abelian ones
  • leading into study of matrix representation of Galois group & classifications
• have tools to prove
  • fundamental theorem of algebra
  • insolvability of quintic polynomials
• mention unsolved problems
  • given finite group, exists Galois extension of $\rationals$ having this group as Galois group?

Fixed fields

for field, $K$, and group of automorphisms, $G$, of $K$, $$ \set{x\in K}{\forall \sigma\in G, x^\sigma = x}\subset K $$ is subfield of $K$, and called fixed field of $G$, denoted by $K^G$

• $K^G$ is subfield of $K$ because for every $x,y\in K^G$
  • $0^\sigma = 0 \Rightarrow 0\in K^G$
  • $(x+y)^\sigma = x^\sigma + y^\sigma = x + y \Rightarrow x+y \in K^G$
  • $(-x)^\sigma = - x^\sigma = - x \Rightarrow -x \in K^G$
  • $1^\sigma = 1 \Rightarrow 1\in K^G$
  • $(xy)^\sigma = x^\sigma y^\sigma = xy \Rightarrow xy\in K^G$
  • $(x^{-1})^\sigma = (x^\sigma)^{-1} = x^{-1} \Rightarrow x^{-1} \in K^G$
  hence, $K^G$ closed under addition & multiplication, and is commutative division ring, thus field
• $0,1\in K^G$, hence $K^G$ contains prime field

Galois extensions and Galois groups

algebraic extension, $K$, of field, $k$, which is normal and separable, said to be Galois (extension of $k$) or Galois over $k$ considering $K$ as embedded in $\algclosure{k}$; for convenience, sometimes say $K/k$ is Galois

for field, $k$ and its Galois extension, $K$, group of automorphisms of $K$ over $k$, called Galois group of $K$ over $k$, denoted by $G(K/k)$, $G_{K/k}$, $\Gal(K/k)$, or (simply) $G$

for field, $k$, separable $f\in k[X]$ with $\deg f \geq 1$, and its splitting field, $K/k$, Galois group of $K$ over $k$ (i.e., $G(K/k)$), called Galois group of $f$ over $k$

for field, $k$, separable $f\in k[X]$ with $\deg f \geq 1$, and its splitting field, $K/k$, $$ f(X) = (X-\alpha_1) \cdots (X-\alpha_n) $$ elements of Galois group of $f$ over $k$, $G$, permute roots of $f$, hence, exists injective homeomorphism of $G$ into $S_n$, i.e., symmetric group on $n$ elements

Fundamental theorem for Galois theory

for finite Galois extension, $K/k$

• map $H \mapsto K^H$ induces isomorphism between set of subgroups of $G(K/k)$ & set of intermediate fields
• subgroup, $H$, of $G(K/k)$, is normal if and only if $K^H/k$ is Galois
• for normal subgroup, $H$, $\sigma\mapsto \restrict{\sigma}{K^H}$ induces isomorphism between $G(K/k)/H$ and $G(K^H/k)$

(illustrated in the figure)

• shall prove step by step

Galois subgroups association with intermediate fields

for Galois extension, $K/k$, and intermediate field, $F$

• $K/F$ is Galois & $K^{G(K/F)} = F$, hence, $K^G = k$
• map $$ F \mapsto G(K/F) $$ induces injective homeomorphism from set of intermediate fields to subgroups of $G$

for Galois extension, $K/k$, and intermediate field, $F$, subgroup, $G(K/F)\subset G(K/k)$, called group associated with $F$, said to belong to $F$

for Galois extension, $K/k$, and two intermediate fields, $F_1$ and $F_2$, $G(K/F_1) \cap G(K/F_2)$ belongs to $F_1F_2$, i.e., $$ G(K/F_1) \cap G(K/F_2) = G(K/F_1F_2) $$

for Galois extension, $K/k$, and two intermediate fields, $F_1$ and $F_2$, smallest subgroup of $G$ containing $G(K/F_1)$ and $G(K/F_2)$ belongs to $F_1\cap F_2$, i.e. $$ \bigcap_{G(K/F_1)\subset H, G(K/F_2)\subset H} \set{H}{H\subset G(K/k)} = G(K/(F_1\cap F_2)) $$

for Galois extension, $K/k$, and two intermediate fields, $F_1$ and $F_2$, $$ F_1\subset F_2 \mbox{ \iaoi\ } G(K/F_2)\subset G(K/F_1) $$

for finite separable field extension, $E/k$, the smallest normal extension of $k$ containing $E$, $K$, $K/k$ is finite Galois and exist only finite number of intermediate fields

for algebraic separable extension, $E/k$, if every element of $E$ has degree no greater than $n$ over $k$ for some $n\geq1$, $E$ is finite over $k$ and $[E:k]\leq n$

(Artin) for field, $K$, finite $\Aut(K)$ of order, $n$, and $k = K^{\Aut(K)}$, $K/k$ is Galois, $G(K/k)= \Aut(K)$, and $[K:k] = n$

for finite Galois extension, $K/k$, every subgroup of $G(K/k)$ belongs to intermediate field

for Galois extension, $K/k$, and intermediate field, $F$,

• $F/k$ is normal extension if and only if $G(K/F)$ is normal subgroup of $G(K/k)$
• if $F/k$ is normal extension, map, $\sigma \mapsto \restrict{\sigma}{F}$, induces homeomorphism of $G(K/k)$ onto $G(F/k)$ of which $G(K/F)$ is kernel, thus $$ G(F/k) \isomorph G(K/k)/G(K/F) $$

Proof for fundamental theorem for Galois theory

• finally, we prove fundamental theorem for Galois theory ()
• assume $K/k$ is finite Galois extension and $H$ is subgroup of $G(K/k)$
  • implies $K^H$ is intermediate field, hence implies $K/K^H$ is Galois, implies $G(K/K^H) = H$, thus, every $H$ is Galois
  • map, $H\mapsto K^H$, induces homeomorphism, $\sigma$, of set of all subgroups of $G(K/k)$ into set of intermediate fields
  • $\sigma$ is injective since for any two subgroups, $H$ and $H'$, of $G(K/k)$, if $K^H=K^{H'}$, then $H=G(K/K^H)=G(K/K^{H'})=H'$
  • $\sigma$ is surjective since for every intermediate field, $F$, implies $K/F$ is Galois, $G(K/F)$ is subgroup of $G(K/k)$, and $K^{G(K/F)}=F$, thus, $\sigma(G(K/F)) = K^{G(K/F)}= F$
  • therefore, $\sigma$ is isomorphism between set of all subgroups of $G(K/k)$ and set of intermediate fields
  • since implies separable extensions are distinguished, $H^K/k$ is separable, thus implies that $K^H/k$ is Galois if and only if $G(K/K^H)$ is normal
  • lastly, implies that if $K^H/k$ is Galois, $G(H^K/k) \isomorph G(K/k) / H$

Abelian and cyclic Galois extensions and groups

Galois extension with abelian Galois group, said to be abelian

Galois extension with cyclic Galois group, said to be cyclic

for Galois extension, $K/k$, and intermediate field, $F$,

• if $K/k$ is abelian, $F/k$ is Galois and abelian
• if $K/k$ is cyclic, $F/k$ is Galois and cyclic

for field, $k$, compositum of all abelian Galois extensions of $k$ in given $\algclosure{k}$, called maximum abelian extension of $k$, denoted by \maxabext{k}

Theorems and corollaries about Galois extensions

for Galois extension, $K/k$, and arbitrary extension, $F/k$, where $K$ and $F$ are subfields of common field,

• $KF / F$ and $K/(K\cap F)$ are Galois extensions
• map $$ \sigma \mapsto \sigma|K $$ induces isomorphism between $G(KF / F)$ and $G(K/(K\cap F))$

theorem illustrated in the figure

for finite Galois extension, $K/k$, and arbitrary extension, $F/k$, where $K$ and $F$ are subfields of common field, $$ [KF:F] \mbox{ divides } [F:k] $$

for Galois extensions, $K_1/k$ and $K_2/k$, where $K_1$ and $K_2$ are subfields of common field,

• $K_1K_2/k$ is Galois extension
• map $$ \sigma \mapsto (\restrict{\sigma}{K_1}, \restrict{\sigma}{K_2}) $$ of $G(K_1K_2/k)$ into $G(K_1/k) \times G(K_2/k)$ is injective; if $K_1\cap K_2=k$, map is isomorphism

theorem illustrated in the figure

for $n$ Galois extensions, $K_i/k$, where $K_1$, , $K_n$ are subfields of common field and $K_{i+1}\cap(K_1\cdots K_i) = k$ for $i=1,\ldots,n-1$,

• $K_1\cdots K_n/k$ is Galois extension
• map $$ \sigma \mapsto (\restrict{\sigma}{K_1}, \ldots, \restrict{\sigma}{K_n}) $$ induces isomorphism of $G(K_1\cdots K_n/k)$ onto $G(K_1/k) \times \cdots \times G(K_n/k)$

for Galois extension, $K/k$, where $G(K/k)$ can be written as $G_1\times \cdots \times G_n$, and $K_1$, , $K_n$, each of which is fixed field of $$ G_1 \times \cdots \times \underbrace{\{e\}}_{i\mathrm{th\ position}} \times \cdots \times G_n $$

• $K_1/k$, , $K_n/k$ are Galois extensions
• $G(K_i/k)=G_i$ for $i=1,\ldots,n$
• $K_{i+1}\cap(K_1\cdots K_i) = k$ for $i=1,\ldots,n-1$
• $K=K_1\cdots K_n$

assume all fields are subfields of common field

• for two abelian Galois extensions, $K/k$ and $L/k$, $KL/k$ is abelian Galois extension
• for abelian Galois extension, $K/k$, and any extension, $E/k$, $KE/E$ is abelian Galois extension
• for abelian Galois extension, $K/k$, and intermediate field, $E$, both $K/E$ and $E/k$ are abelian Galois extensions

Solvable and radical extensions

finite separable extension, $E/k$, such that Galois group of smallest Galois extension, $K/k$, containing $E$ is solvable, said to be solvable

solvable extensions form distinguished class of extensions

finite extension, $F/k$, such that it is separable and exists finite extension, $E/k$, containing $F$ admitting tower decomposition $$ k = E_0 \subset E_1 \subset \cdots \subset E_m = E $$ with $E_{i+1}/E_i$ is obtained by adjoining root of

• unity, or
• $X^n=a$ with $a\in E_i$, and $n$ prime to characteristic, or
• $X_p-X-a$ with $a\in E_i$ if $p$ is positive characteristic

said to be solvable by radicals

separable extension, $E/k$, is solvable by radicals if and only if it is solvable

Applications of Galois theory

general equation of degree, $n$, cannot be solved by radicals for $n\geq5$ (implied by , , , and )

$f\in \complexes[X]$ of degree, $n$, has precisely $n$ roots in $\complexes$ (when counted with multiplicity), hence $\complexes$ is algebraically closed