Table of Contents

Index of Definitions

Index of Theorems

1 Integers

2 Functions

3 Groups

4 Polynomials

5 Rings

6 Fields

7 Structure of Groups

8 Galois Theory

9 Unique Factorization

10 Modules

11 Structure of Noncommutative Rings

12 Ideal Theory of Commutative Rings

List of Theorems




abelian group

action, of a group

algebraic element

algebraic extension

algebraic numbers

alternating group

annihilator, of a module

Artinian module

Artinian ring

ascending central series

associated prime ideal

automorphism, of a group

automorphism, of a ring

bicommutator, of a module

bilinear function


center of a group

centralizer, of an element

characteristic, of a ring

codomain, of a function

commutative ring


completely reducible module

composite number

composition, of functions

composition series, for a group

composition series, for a module

congruence class of integers

congruence, modulo n

congruence, of polynomials

conjugate, of a group element

constructible number


cycle of length k

cyclic group

cyclic module

cyclic subgroup

cyclotomic polynomial

Dedekind domain

degree of a polynomial

degree of an algebraic element

degree of an extension field

derived subgroup

dense subring

dihedral group

disjoint cycles

division ring

divisor, of a polynomial

divisor, of an integer

divisor, of zero

direct product, of groups

direct product, of modules

direct sum, of modules

direct sum, of rings

domain, of a function

equivalence class

equivalence classes defined by a function

equivalence relation

essential submodule

Euclidean domain

Euler's phi-function

even permutation

extension field

factor, of a polynomial

factor, of an integer

factor group

factor ring

faithful module


finite extension field

finite group

finitely generated module

fixed subfield

formal derivative

fractional ideal

free module

Frobenius automorphism


Galois field

Galois group of a polynomial

general linear group

generator, of a cyclic group

greatest common divisor, of integers

greatest common divisor, of polynomials

greatest common divisor, in a principal ideal domain


general linear
of permutations
of quaternions
order of
projective special linear
special linear
group algebra

group ring

holomorph (of the integers mod n)

homomorphism, of groups

homomorphism, of modules

homomorphism, of rings


idempotent element, of a ring

image, of a function

index of a subgroup

injective module

inner automorphism, of a group

integral closure

integral domain

integral extension

integrally closed domain

invariant subfield

inverse function

invertible element, in a ring

irreducible element, in a ring

irreducible polynomial

isomorphism, of groups

isomorphism, of rings

Jacobson radical, of a module

kernel, of a group homomorphism

kernel, of a ring homomorphism

Krull dimension

leading coefficient

least common multiple, of integers

left ideal

Legendre symbol

linear action

localization at a prime ideal

maximal ideal

maximal submodule

minimal polynomial

minimal submodule


Moebius function

monic polynomial

multiple, of an integer

multiplicity, of a root

nil ideal

nil radical

nilpotent element, of a ring

nilpotent ideal

Noetherian module

Noetherian ring

normal extension

normal subgroup

normalizer, of a subgroup

one-to-one function

onto function

odd permutation


order of a group

order of a permutation


partition of a set

perfect extension


permutation group

primary ideal

primitive polynomial

principal left ideal

product, of polynomials

projective module


prime ideal, of a commutative ring

prime ideal, of a noncommutative ring

prime module

prime number

prime ring

primitive ideal

primitive ring

principal ideal

principal ideal domain

quadratic residue


radical, for modules

radical, of an ideal

radical extension

regular element

relatively prime integers

right ideal


ring of differential operators

root of a polynomial

root of unity

semidirect product

semiprime ideal

semiprime ring

semiprimitive ring

semisimple Artinian ring

simple extension

semisimple module

separable polynomial

separable extension

simple group

simple ring

simple extension

simple module

skew field

small submodule

socle of a module

solvable by radicals

split homomorphism

splitting field





Sylow subgroup

symmetric group

tensor product

torsion module

torsionfree module

transcendental element


unique factorization domain

unit, of a ring

von Neumann regular ring

well-ordering principle

zero divisor




An algebraic extension of an algebraic extension is algebraic(6.2.10)

Artin-Wedderburn theorem(11.3.2)

Artin's lemma(8.3.4)

Baer's criterion for injectivity(10.5.9)

Burnside's theorem(7.2.8)

Cauchy's theorem(7.2.10)

Cayley's theorem(3.6.2)

Characteristic of an integral domain(5.2.10)

Characterization of completely reducible modules(10.2.9)

Characterization of completely reducible rings(10.5.6)

Characterization of constructible numbers(6.3.6)

Characterization of Dedekind domains(12.1.6)

Characterization of equations solvable by radicals(8.4.6)

Characterization of finite fields(6.5.2)

Characterization of finite normal separable extensions(8.3.6)

Characterization of free modules(10.2.3)

Characterization of integral elements(12.2.2)

Characterization of internal direct products(7.1.3)

Characterization of invertible functions(2.1.8)

Characterization of the Jacobson radical(11.2.10)

Characterization of linear actions(7.9.5)

Characterization of nilpotent groups(7.8.4)

Characterization of Noetherian modules(10.3.3)

Characterization of normal subgroups(3.8.7)

Characterization of projective modules(10.2.11)

Characterization of semisimple Artinian rings(11.3.4)

Characterization of prime ideals(11.1.3)

Characterization of semidirect products(7.9.6)

Characterization of semiprime ideals(11.1.7)

Characterization of semisimple modules(10.5.3)

Characterization of subgroups(3.2.2)

Characterization of subrings(5.1.3)

Chinese remainder theorem, for integers(1.3.6)

Chinese remainder theorem, for rings(5.7.9)

Class equation(7.2.6)

Class equation (generalized)(7.3.6)

Classification of cyclic groups(3.5.2)

Classification of groups of order less than sixteen

Classification of groups of order pq(7.4.6)

Cohen's theorem(12.4.1)

Computation of Euler's phi-function(1.4.8)

Construction of extension fields(4.4.8)

Correspondence between roots and linear factors(4.1.11)

Dedekind's theorem on reduction modulo p

Properties of Dedekind domains(12.1.4)

Degree of a tower of finite extensions(6.2.4)

DeMoivre's theorem(A.5.2)

The direct product of nilpotent groups is nilpotent(7.8.2)

Disjoint cycles commute(2.3.4)

Division algorithm for integers(1.1.3)

Division algorithm for polynomials(4.2.1)

Eisenstein's irreducibility criterion(4.3.6)

Endomorphisms of indecomposable modules(10.4.6)

Existence of finite fields(6.5.7)

Existence of greatest common divisors (for integers)(1.1.6)

Existence of greatest common divisors (for polynomials)(4.2.4)

Existence of greatest common divisors, in a principal ideal domain(9.1.6)

Existence of irreducible polynomials(6.5.12)

Existence of maximal submodules(10.1.8)

Existence of quotient fields(5.4.4)

Existence of splitting fields(6.4.2)

Existence of tensor products(10.6.3)

Euclidean algorithm for integers

Euclidean algorithm for polynomials(Example 4.2.3)

Euclid's lemma characterizing primes(1.2.5)

Euclid's theorem on the infinitude of primes(1.2.7)

Euler's theorem(1.4.11)

Euler's theorem(Example 3.2.12)

Euler's criterion(6.7.2)

Every Euclidean domain is a principal ideal domain(9.1.2)

Every field of characteristic zero is perfect(8.2.6)

Every finite extension is algebraic(6.2.9)

Every finite separable extension is a simple extension(8.2.8)

Every finite field is perfect(8.2.7)

Every PID is a UFD(9.1.12)

Finite integral domains are fields(5.1.8)

Every finite p-group is solvable(7.6.3)

Finitely generated torsion modules over a PID(10.3.9)

Finitely generated torsionfree modules over a PID(10.7.5)

First isomorphism theorem(7.1.1)

Fitting's lemma for modules(10.4.5)

Frattini's argument(7.8.5)

Fundamental theorem of algebra(8.3.10)

Fundamental theorem of arithmetic(1.2.6)

Fundamental theorem of finitely generated modules over a PID(10.7.5)

Fundamental theorem of Galois theory(8.3.8)

Fundamental theorem of finite abelian groups(7.5.4)

Fundamental homomorphism theorem for groups(3.8.8)

Fundamental homomorphism theorem for rings(5.2.6)

F[x] is a principal ideal domain(4.2.2)

On Galois groups(8.4.3, 8.4.4)

Galois groups of cyclotomic polynomials(8.5.4)

Galois groups over finite fields(8.1.7)

Galois groups and permutations of roots(8.1.4)

Gauss's lemma(4.3.4)

When the group of units modulo n is cyclic(7.5.11)

Hilbert basis theorem(10.3.7)

Hilbert's nullstellensatz(12.4.9)

Hopkin's theorem(11.3.5)

Ideals in the localization of an integral domain(5.8.11)

Impossibility of trisecting an angle(6.3.9)

Incomparability, lying-over, and going up(12.2.9)

Insolvability of the quintic(8.4.8)

Irreducibility of cyclotomic polynomials(8.5.3)

Irreducible ideals are primary(12.3.6)

Irreducible polynomials over R(A.5.7)

Jacobson density theorem(11.3.7)

Jordan-Holder theorem for groups(7.6.10)

Jordan-Holder theorem for modules(10.4.2)

Kronecker's theorem(4.4.8)

Krull's theorem(12.4.6)

Krull-Schmidt theorem(10.4.9)

Lagrange's theorem(3.2.10)

Lasker-Noether decomposition theorem(12.3.10)

Maschke's theorem(10.5.8)

Maximal subgroups in nilpotent groups(7.8.5)

Moebius inversion formula(6.6.6)

The multiplicative group of a finite field is cyclic(6.5.10)

Nakayama's lemma(11.2.8)

The nil radical is nilpotent (in Noetherian rings)(12.4.3)

Number of irreducible polynomials over a finite field(6.6.9)

Number of roots of a polynomial(4.1.12)

Order of a permutation(2.3.8)

Order of the Galois group of a polynomial(8.1.6)

Partial fractions(Example 4204)

Every p-group is abelian(7.2.9)

Every permutation is a product of disjoint cycles(2.3.5)

The polynomial ring over a UFD is a UFD(9.2.6)

The ring of power series is Noetherian(12.4.2)

Prime and maximal ideals(5.3.9)

Prime ideals in a principal ideal domain(5.3.10)

Generalized principal ideal theorem(12.4.7)

Quadratic reciprocity law(6.7.3)

Rational roots(4.3.1)

Remainder theorem(4.1.9)

Schur's lemma(10.1.11)

Second isomorphism theorem(7.1.2)

Simplicity of PSL(2,F)(7.7.9)

Simplicity of the alternating group(7.7.4)

The smallest nonabelian simple group(7.10.7)

On solvable groups(7.6.7, 7.6.8)

Splitting fields are unique(6.4.5)

Structure of simple extensions(6.1.6)

Subgroups of cyclic groups(3.5.1)

Sylow's theorems(7.4.1, 7.4.4)

When the symmetric group is solvable(7.7.2)

Unique factorization of integers(1.2.6)

Unique factorization of polynomials(4.2.9)

Wedderburn's theorem(8.5.6)