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\( \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \newcommand{\T}{{\mathbb T}} \newcommand{\F}{{\mathbb F}} \newcommand{\PP}{{\mathbb P}} \newcommand{\HH}{{\mathbb H}} \newcommand{\compose}{\circ} \newcommand{\bolda}{{\mathbf a}} \newcommand{\boldb}{{\mathbf b}} \newcommand{\boldc}{{\mathbf c}} \newcommand{\boldd}{{\mathbf d}} \newcommand{\bolde}{{\mathbf e}} \newcommand{\boldi}{{\mathbf i}} \newcommand{\boldj}{{\mathbf j}} \newcommand{\boldk}{{\mathbf k}} \newcommand{\boldn}{{\mathbf n}} \newcommand{\boldp}{{\mathbf p}} \newcommand{\boldq}{{\mathbf q}} \newcommand{\boldr}{{\mathbf r}} \newcommand{\bolds}{{\mathbf s}} \newcommand{\boldt}{{\mathbf t}} \newcommand{\boldu}{{\mathbf u}} \newcommand{\boldv}{{\mathbf v}} \newcommand{\boldw}{{\mathbf w}} \newcommand{\boldx}{{\mathbf x}} \newcommand{\boldy}{{\mathbf y}} \newcommand{\boldz}{{\mathbf z}} \newcommand{\boldzero}{{\mathbf 0}} \newcommand{\boldmod}{\boldsymbol{ \bmod }} \newcommand{\boldT}{{\mathbf T}} \newcommand{\boldN}{{\mathbf N}} \newcommand{\boldB}{{\mathbf B}} \newcommand{\boldF}{{\mathbf F}} \newcommand{\boldS}{{\mathbf S}} \newcommand{\boldE}{{\mathbf E}} \newcommand{\boldG}{{\mathbf G}} \newcommand{\boldK}{{\mathbf K}} \newcommand{\boldL}{{\mathbf L}} \DeclareMathOperator{\ch}{char} \DeclareMathOperator{\lns}{lns} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\NS}{null} \DeclareMathOperator{\RS}{row} \DeclareMathOperator{\CS}{col} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\nullity}{nullity} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Arg}{Arg} \DeclareMathOperator{\Log}{Log} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Arcsin}{Arcsin} \DeclareMathOperator{\Arccos}{Arccos} \DeclareMathOperator{\Arctan}{Arctan} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\Aff}{Aff} \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\mdeg}{mdeg} \DeclareMathOperator{\Lt}{Lt} \DeclareMathOperator{\Lc}{Lc} \DeclareMathOperator{\disc}{disc} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\adj}{adj} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\curl}{curl} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\diver}{div} \DeclareMathOperator{\flux}{flux} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Isom}{Isom} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\ML}{M} \DeclareMathOperator{\Syl}{Syl} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\ab}{ab} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\len}{len} \DeclareMathOperator{\proj}{proj} \newcommand{\surjects}{\twoheadrightarrow} \newcommand{\injects}{\hookrightarrow} \newcommand{\bijects}{\leftrightarrow} \newcommand{\isomto}{\overset{\sim}{\rightarrow}} \newcommand{\floor}[1]{\lfloor#1\rfloor} \newcommand{\ceiling}[1]{\left\lceil#1\right\rceil} \newcommand{\mclass}[2][m]{[#2]_{#1}} \newcommand{\val}[2][]{\left\lvert #2\right\rvert_{#1}} \newcommand{\abs}[2][]{\left\lvert #2\right\rvert_{#1}} \newcommand{\valuation}[2][]{\left\lvert #2\right\rvert_{#1}} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\anpoly}{a_nx^n+a_{n-1}x^{n-1}\cdots +a_1x+a_0} \newcommand{\anmonic}{x^n+a_{n-1}x^{n-1}\cdots +a_1x+a_0} \newcommand{\bmpoly}{b_mx^m+b_{m-1}x^{m-1}\cdots +b_1x+b_0} \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}} \newcommand{\normalin}{\trianglelefteq} \newcommand{\angvec}[1]{\langle #1\rangle} \newcommand{\varpoly}[2]{#1_{#2}x^{#2}+#1_{#2-1}x^{#2-1}\cdots +#1_1x+#1_0} \newcommand{\varpower}[1][a]{#1_0+#1_1x+#1_1x^2+\cdots} \newcommand{\limasto}[2][x]{\lim_{#1\rightarrow #2}} \newcommand{\abcdmatrix}[4]{\begin{bmatrix}#1\amp #2\\ #3\amp #4 \end{bmatrix} } \newenvironment{amatrix}[1][ccc|c]{\left[\begin{array}{#1}}{\end{array}\right]} \newenvironment{linsys}[2][m]{ \begin{array}[#1]{@{}*{#2}{rc}r@{}} }{ \end{array}} \newcommand{\eqsys}{\begin{array}{rcrcrcr} a_{11}x_{1}\amp +\amp a_{12}x_{2}\amp +\cdots+\amp a_{1n}x_{n}\amp =\amp b_1\\ a_{21}x_{1}\amp +\amp a_{22}x_{2}\amp +\cdots+\amp a_{2n}x_{n}\amp =\amp b_2\\ \amp \vdots\amp \amp \vdots \amp \amp \vdots \amp \\ a_{m1}x_{1}\amp +\amp a_{m2}x_{2}\amp +\cdots +\amp a_{mn}x_{n}\amp =\amp b_m \end{array} } \newcommand{\numeqsys}{\begin{array}{rrcrcrcr} e_1:\amp a_{11}x_{1}\amp +\amp a_{12}x_{2}\amp +\cdots+\amp a_{1n}x_{n}\amp =\amp b_1\\ e_2: \amp a_{21}x_{1}\amp +\amp a_{22}x_{2}\amp +\cdots+\amp a_{2n}x_{n}\amp =\amp b_2\\ \amp \vdots\amp \amp \vdots \amp \amp \vdots \amp \\ e_m: \amp a_{m1}x_{1}\amp +\amp a_{m2}x_{2}\amp +\cdots +\amp a_{mn}x_{n}\amp =\amp b_m \end{array} } \newcommand{\homsys}{\begin{array}{rcrcrcr} a_{11}x_{1}\amp +\amp a_{12}x_{2}\amp +\cdots+\amp a_{1n}x_{n}\amp =\amp 0\\ a_{21}x_{1}\amp +\amp a_{22}x_{2}\amp +\cdots+\amp a_{2n}x_{n}\amp =\amp 0\\ \amp \vdots\amp \amp \vdots \amp \amp \vdots \amp \\ a_{m1}x_{1}\amp +\amp a_{m2}x_{2}\amp +\cdots +\amp a_{mn}x_{n}\amp =\amp 0 \end{array} } \newcommand{\vareqsys}[4]{ \begin{array}{ccccccc} #3_{11}x_{1}\amp +\amp #3_{12}x_{2}\amp +\cdots+\amp #3_{1#2}x_{#2}\amp =\amp #4_1\\ #3_{21}x_{1}\amp +\amp #3_{22}x_{2}\amp +\cdots+\amp #3_{2#2}x_{#2}\amp =\amp #4_2\\ \vdots \amp \amp \vdots \amp \amp \vdots \amp =\amp \\ #3_{#1 1}x_{1}\amp +\amp #3_{#1 2}x_{2}\amp +\cdots +\amp #3_{#1 #2}x_{#2}\amp =\amp #4_{#1} \end{array} } \newcommand{\genmatrix}[1][a]{ \begin{bmatrix} #1_{11} \amp #1_{12} \amp \cdots \amp #1_{1n} \\ #1_{21} \amp #1_{22} \amp 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Appendix B Definitions

0.1 Sets and functions

Definition 0.1.1 Sets

Definition 0.1.2 Set equality

Definition 0.1.3 Set inclusion (subsets)

Definition 0.1.5 Set-builder notation

Definition 0.1.8 Union, intersection, difference, and complement

Definition 0.1.9 Common mathematical sets

Definition 0.1.10 Power set

Definition 0.1.12 Functions

Definition 0.1.16 Function equality

Definition 0.1.17 Image of a set

Definition 0.1.18 Preimage of set

Definition 0.1.19 Injective, surjective, bijective

Definition 0.1.21 Function composition

Definition 0.1.22 Identity and inverse functions

Definition 0.1.24 Tuple

Definition 0.1.28 \(n\)-tuples

Definition 0.1.31 Infinite sequence

Definition 0.1.33 Cartesian product

Definition 0.1.34 Cardinality

0.2 Logic

Definition 0.2.1 Logical operators

Definition 0.2.5 Logical quantifiers

1.1 Groups

Definition 1.1.1 Groups

Definition 1.1.3 Abelian group

Definition 1.1.5 Product group

Definition 1.1.9 Group exponentiation

Definition 1.1.10 Group element order

1.2 Modular arithmetic

Definition 1.2.1 Divisibilty

Definition 1.2.2 Congruence modulo \(n\)

Definition 1.2.6 Least residue modulo \(n\)

Definition 1.2.14 Multiplicative units in \(\Z/n\Z\)

Definition 1.2.17 Greatest common divisor

1.3 Matrix groups

Definition 1.3.1 Invertible matrix

1.4 Dihedral groups

Definition 1.4.1 Isometry

1.5 Permutations

Definition 1.5.1 Permutations

Notation 1.5.2 Table notation

Definition 1.5.4 Cycles

1.6 Homomorphisms

Definition 1.6.1 Homomorphism

Definition 1.6.6 Isomorphism

1.7 Group actions

Definition 1.7.1 Group action

Definition 1.7.4 Permutation representation

1.8 Subgroups

Definition 1.8.1 Subgroup

Definition 1.8.5 Cyclic groups

Definition 1.8.9 Kernel of homomorphism

Definition 1.8.12 Centralizer, normalizer, center

1.10 Subgroup lattice

Definition 1.10.2 Subgroup generated by a set

1.11 Cosets and Lagrange’s theorem

Definition 1.11.1 Group operations on subsets

Definition 1.11.2 Coset and coset space

Notation 1.11.3 Cosets are left cosets by default

Definition 1.11.9 Index of subgroup

1.12 Quotient groups

Definition 1.12.1 Normal subgroup

1.14 Isomorphism theorems: second and third

Notation 1.14.6 Images under quotients

1.16 Alternating subgroup

Definition 1.16.1 Transposition

Definition 1.16.3 Even and odd permutations

1.17 Group actions: stabilizer

Definition 1.17.1 G-sets

Definition 1.17.2 Stabilizers and kernels of actions

Definition 1.17.4 Permutation representation and faithfulness

1.18 Group actions: orbits

Definition 1.18.1 Orbits

Definition 1.18.2 Transitive group action

Definition 1.18.7 Fix sets

1.19 Class equation

Definition 1.19.1 Conjugacy class

Definition 1.19.3 Class equation

Definition 1.19.8 Partition of \(n\)

1.20 Sylow theorems: intro

Definition 1.20.1 Order at \(p\)

Definition 1.20.3 Sylow subgroups

Definition 1.20.8 \(p\)-group

1.23 Direct products

Definition 1.23.2 Interal direct product

Definition 1.23.5 Torsion subgroups

1.24 Semidirect products

Definition 1.24.2 Homomorphism set

2.1 Rings

Definition 2.1.1 Ring

Definition 2.1.7 Subring

Definition 2.1.8 Center of ring

2.2 Subrings, units, zero divisors

Definition 2.2.1 Units and multiplicative inverses

Definition 2.2.4 Division rings and fields

Definition 2.2.7 Zero divisors and integral domains

Definition 2.2.11 Nilpotents and idempotents

2.4 Hamilton quaternion rings

Definition 2.4.1 Quaternion conjugation and norm

2.5 Ring homomorphisms and ideals

Definition 2.5.1 Ring homomorphism

Definition 2.5.9 Change of coefficients, reduction, evaluation

Definition 2.5.13 Ideal

Definition 2.5.16 Generated ideals

Definition 2.5.20 Characteristic of a ring

Definition 2.5.24 Simple ring

Definition 2.5.26 \(R\)-algebra

2.7 Prime and maximal ideals

Definition 2.7.1 Maximal ideal

Definition 2.7.5 Prime ideal

Definition 2.7.9 Partial order

2.8 Case study: polynomial rings

Definition 2.8.2 Polynomial division with remainder

Definition 2.8.5 Principal ideal domain

Definition 2.8.7 Divisibility

Definition 2.8.8 Prime and irreducible elements

Definition 2.8.18 Root of a polynomial

2.9 Chinese remainder theorem

Definition 2.9.1 Coprime ideals

Definition 2.9.8 Euler’s totient function

2.10 Localizations and fraction fields

Definition 2.10.1 Localization

Definition 2.10.7 Fraction field

Notation 2.10.13 Special types of localizations

Definition 2.10.19 Primitive polynomial in \(\Z[x]\)

2.11 Euclidean domains

Definition 2.11.1 Euclidean domain

Definition 2.11.2 Norm function on \(\C\)

Definition 2.11.5 Greatest common divisor and least common multiple

2.12 PIDs and UFDs

Definition 2.12.4 Unique factorization domain

Definition 2.12.8 Valuation at irreducible

2.13 Gaussian integers

Definition 2.13.1 Quadratic ring of integers

Definition 2.13.6 Lying over

2.14 Gauss’s lemma

Definition 2.14.1 Primitive polynomials

2.15 Multivariate polynomial rings and monoid algebras

Definition 2.15.2 Monomials and multivariate indices

Definition 2.15.5 Noetherian ring

Definition 2.15.9 Total degree

Definition 2.15.12 Monoid

2.17 Modules

Definition 2.17.1 Modules

Definition 2.17.5 Submodule

Definition 2.17.7 Module homomorphism

Definition 2.17.9 \(R\)-linear combination

Definition 2.17.10 Generated submodule

2.18 Direct sums and free modules

Definition 2.18.4 Finitely generated module

Definition 2.18.6 Linear independence and span

2.19 PID-module structure theorems: statements

Definition 2.19.1 Annihilators and torsion elements

Definition 2.19.3 Betti number and invariant factors

Definition 2.19.6 \(\alpha\)-torsion and \(\pi\)-primary submodules

Definition 2.19.8 Elementary divisors

2.20 PID-module structure theorems: proofs

Definition 2.20.4 Aligned basis

Definition 2.20.10 Standard matrix representation

Definition 2.20.15 Elementary operations

2.21 PID-module structure theorems: Jordan canonical forms

Definition 2.21.2 Minimal and characteristic polynomial