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
Copyright 1996, 160 pages
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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)
- Quadratic reciprocity law(6.7.3)
- 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)
- Characterization of the Jacobson radical(11.2.10)
- 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)
- Lasker-Noether decomposition theorem(12.3.10)
- Cohen's theorem(12.4.1)
- The ring of power series is Noetherian(12.4.2)
- The nil radical is nilpotent (in Noetherian rings)(12.4.3)
- Krull's theorem(12.4.6)
- Generalized principal ideal theorem(12.4.7)
- Hilbert's nullstellensatz(12.4.9)
- DeMoivre's theorem(A.5.2)
- Irreducible polynomials over R(A.5.7)
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