ABSTRACT ALGEBRA ON LINE
This site contains many of the definitions and theorems
from the area of mathematics generally called abstract algebra.
It is intended for undergraduate students
taking an abstract algebra class at the junior/senior level,
as well as for students taking their first graduate algebra course.
It is based on the books
Abstract Algebra,
by John A. Beachy and William D. Blair, and
Abstract Algebra II,
by John A. Beachy.
The site is organized by chapter.
The page containing the Table of Contents
also contains an index of definitions and theorems,
which can be searched for detailed references on subject area pages.
Topics from the first volume are marked by the symbol
and those from the second volume by the symbol
.
To make use of this site as a reference, please continue on to the
Table of Contents.
 TABLE OF CONTENTS (No frames)
TABLE OF CONTENTS (Frames version)
Interested students may also wish to refer to a closely related site
that includes solved problems: the

OnLine Study Guide for Abstract Algebra.
REFERENCES
Abstract Algebra,
Second Edition,
by
John A. Beachy
and
William D. Blair
ISBN 0881338664,
© 1996, 427 pages
Waveland Press,
P.O. Box 400,
Prospect Heights, Illinois, 60070,
Tel. 847 / 6340081
Abstract Algebra II
This set of lecture notes was expanded into the following text.
Introductory Lectures on Rings and Modules,
by
John A. Beachy
ISBN 0521644070,
© 1999, 238 pages
Cambridge University Press,
London Mathematical Society Student Texts #47
In addition to the Table of Contents,
this page contains an index of definitions and theorems,
so it can be searched for detailed references on subject area pages.
Topics from the first volume are marked by the symbol
and those from the second volume by the symbol
.
Click
here
for the version with frames.
The site is maintained by John Beachy as a service to students.
email:
jbeachy@niu.edu

John Beachy's
homepage

About this document
TABLE OF CONTENTS

Integers

Functions

Groups

Basic group theory

Factor groups and homomorphisms

Some group multiplication tables

Polynomials

Rings

Commutative rings; integral domains

Localization, noncommutative examples

Fields

Structure of Groups

Sylow theorems; abelian groups; solvable groups

Nilpotent groups; groups of small order

Galois Theory

Unique Factorization

Modules

Sums and products; chain conditions

Composition series; tensor products; modules over a PID

Structure of Noncommutative Rings

Ideal Theory of Commutative Rings
INDEX

Index of Definitions

Index of Theorems

List of Theorems
Index of Definitions
 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
 bimodule
 center of a group
 centralizer, of an element
 characteristic, of a ring
 codomain, of a function
 commutative ring
 commutator
 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
 coset
 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 phifunction
 even permutation
 extension field
 factor, of a polynomial
 factor, of an integer
 factor group
 factor ring
 faithful module
 field
 finite extension field
 finite group
 finitely generated module
 fixed subfield
 formal derivative
 fractional ideal
 free module
 Frobenius automorphism
 function
 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
 group
 abelian
 alternating
 cyclic
 dihedral
 finite
 general linear
 nilpotent
 of permutations
 of quaternions
 order of
 projective special linear
 simple
 solvable
 special linear
 symmetric
 transitive
 group algebra
 group ring
 holomorph (of the integers mod n)
 homomorphism, of groups
 homomorphism, of modules
 homomorphism, of rings
 ideal
 idempotent element, of a ring
 image, of a function
 index of a subgroup
 injective module
 inner automorphism, of a group
 integer
 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
 module
 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
 onetoone function
 onto function
 odd permutation
 orbit
 order of a group
 order of a permutation
 pgroup
 partition of a set
 perfect extension
 permutation
 permutation group
 primary ideal
 primitive polynomial
 principal left ideal
 product, of polynomials
 projective module
 polynomial
 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
 quaternions
 radical, for modules
 radical, of an ideal
 radical extension
 regular element
 relatively prime integers
 right ideal
 ring
 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
 stabilizer
 subfield
 subgroup
 subring
 Sylow subgroup
 symmetric group
 tensor product
 torsion module
 torsionfree module
 transcendental element
 transposition
 unique factorization domain
 unit, of a ring
 von Neumann regular ring
 wellordering principle
 zero divisor
Index of Theorems
 An algebraic extension of an algebraic extension is algebraic(6.2.10)
 ArtinWedderburn 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 phifunction(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 pgroup 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, lyingover, 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)
 JordanHolder theorem for groups(7.6.10)
 JordanHolder theorem for modules(10.4.2)
 Kronecker's theorem(4.4.8)
 Krull's theorem(12.4.6)
 KrullSchmidt theorem(10.4.9)
 Lagrange's theorem(3.2.10)
 LaskerNoether 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 pgroup 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)
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