- 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
- commutator
- 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
- coset
- 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
- field
- finite extension field
- finite group
- fixed subfield
- formal derivative
- 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
- of permutations
- of quaternions
- order of
- projective special linear
- simple
- solvable
- special linear
- symmetric
- transitive

- homomorphism, of groups
- homomorphism, of rings
- ideal
- idempotent element, of a ring
- image, of a function
- index of a subgroup
- inner automorphism, of a group
- integer
- 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
- orbit
- order of a group
- order of a permutation
- p-group
- partition of a set
- perfect extension
- permutation
- permutation group
- primitive polynomial
- product, of polynomials
- polynomial
- 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
- stabilizer
- subfield
- subgroup
- subring
- Sylow subgroup
- symmetric group
- transcendental element
- transposition
- unique factorization domain
- unit, of a ring
- well-ordering principle

- 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
(A.5.7)**R** - 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)