Index of Definitions

abelian group
action, of a group
algebraic element
algebraic extension
algebraic numbers
alternating group
associative operation
automorphism, of a group
automorphism, of a ring

binary operation

center of a group
centralizer, of an element
characteristic, of a ring
codomain, of a function
commutative ring
composite number
composition, of functions
composition series, for a group
congruence class of integers
congruence, modulo n
congruence, of polynomials
conjugate, of a group element
constructible number
cycle of length k
cyclic group
cyclic subgroup
cyclotomic polynomial

degree of a polynomial
degree of an algebraic element
degree of an extension field
derived subgroup
dihedral group
direct product, of groups
direct sum, of rings
disjoint cycles
divisor, of a polynomial
divisor, of an integer
divisor, of zero
domain, of a function

equivalence class
equivalence classes defined by a function
equivalence relation
Euclidean domain
Euler's phi-function
even permutation
extension field

factor, of a polynomial
factor, of an integer
factor group
factor ring
finite extension field
finite group
fixed subfield
formal derivative
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

homomorphism, of groups
homomorphism, of rings

idempotent element, of a ring
image, of a function
index of a subgroup
inner automorphism, of a group
integral domain
invariant subfield
inverse function
irreducible element, in a ring
irreducible polynomial
isomorphism, of groups
isomorphism, of rings

kernel, of a group homomorphism
kernel, of a ring homomorphism

leading coefficient
least common multiple, of integers
Legendre symbol

maximal ideal
minimal polynomial
Moebius function
monic polynomial
multiple, of an integer
multiplicity, of a root

nilpotent element, of a ring
normal extension
normal subgroup
normalizer, of a subgroup

odd permutation
one-to-one function
onto function
operation, binary
operation, associative
order of a group
order of a permutation

partition of a set
perfect extension
permutation group
primitive polynomial
product, of polynomials
prime ideal, of a commutative ring
prime number
principal ideal
principal ideal domain

quadratic residue

radical extension
relatively prime integers
root of a polynomial
root of unity

simple extension
separable polynomial
separable extension
simple group
simple extension
solvable by radicals
splitting field
Sylow subgroup
symmetric group

transcendental element

unique factorization domain
unit, of a ring

well-ordering principle

Index of Theorems

An algebraic extension of an algebraic extension is algebraic(6.2.10)
Artin's lemma(8.3.4)

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 constructible numbers(6.3.6)
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 internal direct products(7.1.3)
Characterization of invertible functions(2.1.8)
Characterization of normal subgroups(3.8.7)
Characterization of subgroups(3.2.2)
Characterization of subrings(5.1.3)
Chinese remainder theorem, for integers(1.3.6)
Class equation(7.2.6)
Class equation (generalized)(7.3.6)
Classification of cyclic groups(3.5.2)
Classification of groups of order pq(7.4.6)
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
Degree of a tower of finite extensions(6.2.4)
DeMoivre's theorem(A.5.2)
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)
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 quotient fields(5.4.4)
Existence of splitting fields(6.4.2)
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)

First isomorphism theorem(7.1.1)
Fundamental theorem of algebra(8.3.10)
Fundamental theorem of arithmetic(1.2.6)
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)

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)
Group of units modulo n - when it is cyclic(7.5.11)

Impossibility of trisecting an angle(6.3.9)
Insolvability of the quintic(8.4.8)
Irreducibility of cyclotomic polynomials(8.5.3)
Irreducible polynomials over R(A.5.7)

Jordan-Holder theorem for groups(7.6.10)

Kronecker's theorem(4.4.8)

Lagrange's theorem(3.2.10)

Moebius inversion formula(6.6.6)
The multiplicative group of a finite field is cyclic(6.5.10)

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)
Prime and maximal ideals(5.3.9)
Prime ideals in a principal ideal domain(5.3.10)

Quadratic reciprocity law(6.7.3)

Rational roots(4.3.1)
Remainder theorem(4.1.9)

Second isomorphism theorem(7.1.2)
Simplicity of PSL(2,F)(7.7.9)
Simplicity of the alternating group(7.7.4)
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)