## List of Theorems

This page contains a list of the major results in the following books.

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

Abstract Algebra II, by John A. Beachy

#### List of Theorems

Division algorithm for integers(1.1.3)
Existence of greatest common divisors (for integers)(1.1.6)
Euclidean algorithm for integers
Euclid's lemma characterizing primes(1.2.5)
Fundamental theorem of arithmetic(1.2.6)
Euclid's theorem on the infinitude of primes(1.2.7)
Chinese remainder theorem for integers(1.3.6)
Computation of Euler's phi-function(1.4.8)
Euler's theorem(1.4.11)
Fermat's "little" theorem(1.4.12)
Characterization of invertible functions(2.1.8)
Disjoint cycles commute(2.3.4)
Every permutation is a product of disjoint cycles(2.3.5)
Order of a permutation(2.3.8)
Characterization of subgroups(3.2.2)
Lagrange's theorem(3.2.10)
Subgroups of cyclic groups(3.5.1)
Classification of cyclic groups(3.5.2)
Cayley's theorem(3.6.2)
Characterization of normal subgroups(3.8.7)
Fundamental homomorphism theorem for groups(3.8.8)
Remainder theorem(4.1.9)
Correspondence between roots and linear factors(4.1.11)
Number of roots of a polynomial(4.1.12)
Division algorithm for polynomials(4.2.1)
F[x] is a principal ideal domain(4.2.2)
Euclidean algorithm for polynomials(Example 4.2.3)
Partial fractions(Example 4.2.4)
Existence of greatest common divisors (for polynomials)(4.2.4)
Unique factorization of polynomials(4.2.9)
Rational roots(4.3.1)
Gauss's lemma(4.3.4)
Eisenstein's irreducibility criterion(4.3.6)
Kronecker's theorem(4.4.8)
Characterization of subrings(5.1.3)
Finite integral domains are fields(5.1.8)
Fundamental homomorphism theorem for rings(5.2.6)
Characteristic of an integral domain(5.2.10)
Prime and maximal ideals(5.3.9)
Prime ideals in a principal ideal domain(5.3.10)
Existence of quotient fields(5.4.4)
Chinese remainder theorem, for rings(5.7.9)
Ideals in the localization of an integral domain(5.8.11)
Structure of simple extensions(6.1.6)
Degree of a tower of finite extensions(6.2.4)
Every finite extension is algebraic(6.2.9)
An algebraic extension of an algebraic extension is algebraic(6.2.10)
Characterization of constructible numbers(6.3.6)
Impossibility of trisecting an angle(6.3.9)
Existence of splitting fields(6.4.2)
Splitting fields are unique(6.4.5)
Characterization of finite fields(6.5.2)
Existence of finite fields(6.5.8)
The multiplicative group of a finite field is cyclic(6.5.10)
Existence of irreducible polynomials(6.5.12)
Moebius inversion formula(6.6.6)
Number of irreducible polynomials over a finite field(6.6.9)
Euler's criterion(6.7.2)
First isomorphism theorem(7.1.1)
Second isomorphism theorem(7.1.2)
Characterization of internal direct products(7.1.3)
Class equation(7.2.6)
Burnside's theorem(7.2.8)
Every p-group is abelian(7.2.9)
Cauchy's theorem(7.2.10)
Class equation (generalized)(7.3.6)
Sylow's theorems(7.4.1, 7.4.4)
Classification of groups of order pq(7.4.6)
Fundamental theorem of finite abelian groups(7.5.4)
When the group of units modulo n is cyclic(7.5.11)
Every finite p-group is solvable(7.6.3)
On solvable groups(7.6.7, 7.6.8)
Jordan-Holder theorem(7.6.10)
When the symmetric group is solvable(7.7.2)
Simplicity of the alternating group(7.7.4)
Simplicity of PSL(2,F)(7.7.9)
The direct product of nilpotent groups is nilpotent(7.8.2)
Characterization of nilpotent groups(7.8.4)
Frattini's argument(7.8.5)
Maximal subgroups in nilpotent groups(7.8.5)
Characterization of linear actions(7.9.5)
Characterization of semidirect products(7.9.6)
Classification of groups of order less than sixteen
The smallest nonabelian simple group(7.10.7)
Galois groups and permutations of roots(8.1.4)
Order of the Galois group of a polynomial(8.1.6)
Galois groups over finite fields(8.1.7)
Every field of characteristic zero is perfect(8.2.6)
Every finite field is perfect(8.2.7)
Every finite separable extension is a simple extension(8.2.8)
Artin's lemma(8.3.4)
Characterization of finite normal separable extensions(8.3.6)
Fundamental theorem of Galois theory(8.3.8)
Fundamental theorem of algebra(8.3.10)
On Galois groups(8.4.3, 8.4.4)
Characterization of equations solvable by radicals(8.4.6)
Insolvability of the quintic(8.4.8)
Irreducibility of cyclotomic polynomials(8.5.3)
Galois groups of cyclotomic polynomials(8.5.4)
Wedderburn's theorem(8.5.6)
Dedekind's theorem on reduction modulo p
Every Euclidean domain is a principal ideal domain(9.1.2)
Existence of greatest common divisors, in a principal ideal domain(9.1.6)
Every PID is a UFD(9.1.12)
The polynomial ring over a UFD is a UFD(9.2.6)
Existence of maximal submodules(10.1.8)
Schur's lemma(10.1.11)
Characterization of free modules(10.2.3)
Characterization of completely reducible modules(10.2.9)
Characterization of projective modules(10.2.11)
Characterization of Noetherian modules(10.3.3)
Hilbert basis theorem(10.3.7)
Finitely generated torsion modules over a PID(10.3.9)
Jordan-Holder theorem for modules(10.4.2)
Fitting's lemma for modules(10.4.5)
Endomorphisms of indecomposable modules(10.4.6)
Krull-Schmidt theorem(10.4.9)
Characterization of semisimple modules(10.5.3)
Maschke's theorem(10.5.8)
Baer's criterion for injectivity(10.5.9)
Existence of tensor products(10.6.3)
Finitely generated torsionfree modules over a PID(10.7.5)
Fundamental theorem of finitely generated modules over a PID(10.7.5)
Characterizaton of prime ideals(11.1.3)
Characterizaton of semiprime ideals(11.1.7)
Nakayama's lemma(11.2.8)
Artin-Wedderburn theorem(11.3.2)
Characterization of semisimple Artinian rings(11.3.4)
Hopkin's theorem(11.3.5)
Jacobson density theorem(11.3.7)
Properties of Dedekind domains(12.1.4)
Characterization of Dedekind domains(12.1.5)
Incomparability, lying-over, and going up(12.2.9)
Irreducible ideals are primary(12.3.6)