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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 (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.





Abstract AlgebraSecond Edition, by John A. Beachy and William D. Blair 
ISBN 0-88133-866-4, © 1996, 427 pages 
Waveland Press, P.O. Box 400, Prospect Heights, Illinois, 60070, Tel. 847 / 634-0081

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 0-521-64407-0, © 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: beachy@math.niu.edu | John Beachy's homepage | About this document








 Basic group theory
 Factor groups and homomorphisms
 Some group multiplication tables




 Commutative rings; integral domains
 Localization, noncommutative examples




 Structure of Groups
 Sylow theorems; abelian groups; solvable groups
 Nilpotent groups; groups of small order


 Galois Theory


 Unique Factorization


 Sums and products; chain conditions
 Composition series; tensor products; modules over a PID


 Structure of Noncommutative Rings


 Ideal Theory of Commutative Rings




 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
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

Index of Theorems


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)

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