Index
Abel, N.H. (1802–1829)
abelian group
finite p-groups
fundamental theorem
primary decomposition theorem
absolute value
absolute value of a complex number
action of a mapping
additive notation
al-Khowarizmi, M. (c.825)
Alexandroff, P.S. (1896–1982)
algebra
regular representation
algebraic closure
algebraic element
algebraic numbers
algebraically closed field
alternating group
alternating polynomial
annihilator
in a module
in a ring
annihilator ideal
Archimedes of Syracuse (272–212 BC)
Artin, E. (1898–1962)
artinian, left
associative law
for composition
general
automorphism
Frobenius
inner
of rings
automorphism group
axiom of choice
axiomatic method
axioms
Bézout domain
Bézout, E. (1730–1783)
basis
of a module
of a vector space
standard
Bass, H. (1932–)
BCH codes
bijection
bijection theorem
bijective mapping
binary operation
associative
closed under
commutative
componentwise
unity for
binomial
coefficient
theorem
bits
Boole, G. (1815–1864)
boolean ring
Brauer's lemma
Brauer, R. (1901–1977)
Burnside's lemma
Burnside, W. (1852–1927)
Burnside-Wielandt theorem
cancellation
Cantor, G. (1845–1918)
Cantor, G. (1845–1918)
Cardano, G. (1501–1576)
cartesian product
casting out nines
Cauchy's theorem
Cauchy, A.L. (1789–1857)
Cauchy-Frobenius lemma
Cayley table
Cayley's theorem
extended
Cayley, A. (1821–1895)
center
of a group
of a ring
central series
central series of a group
ascending
descending
centralizer
chain condition
ascending
descending
on principal ideals
character of a group
Dedekind's lemma
independent
characteristic
characteristic of a ring
characteristic subgroup
Chevalley, C. (1909–1984)
Chinese remainder theorem
circle group
class equation
code
(n,k)-code
BCH
binary
check bits
coset leader for
cyclic
error correction
error detection
Hamming (7,4)
idempotent generator
matrix description
matrix generator
maximum likelihood decoding
minimal generator
nearest neighbor
orthogonality theorem
parity check
perfect
polynomial/word form
standard array for
standard generator matrix
syndrome
syndrome decoding
words
codomain of a mapping
Cohen, P. (1934–)
combinatorics application
commutative law
commutative ring
simple
commutator
subgroup
complemented module
completing the square
complex number
absolute value/modulus of
conjugate of
DeMoivre's theorem
distance formula
exponential form
inverse of
modulus
multiplication rule
operations
polar form of
real/imaginary part
roots of unity
triangle inequality
complex plane
imaginary axis
real axis
component of an n-tuple
componentwise operation
composite of mappings
composition factors
composition length
composition series
group
module
conclusion
congruence modulo n
conjugacy class
conjugate elements
conjugate in quadratic integers
conjugate of a complex number
conjugate subgroups
constructible number
containment
proper
contradiction
contrapositive
converse
convolution
core of a subgroup
corner ring
correspondence theorem
coset
double
left/right
map
coset decoding
coset leader
coset map
counterexample
cubic formula
cubic polynomial
cycle
decomposition theorem
length
structure
cycle index
cyclic codes
cyclic group
fundamental theorem
of order n
cyclic subgroup
generator of
cyclotomic polynomial
D'Alembert, J. LeR. (1717–1783)
De Morgan, A. (1806–1871)
Dedekind's lemma
Dedekind, R. (1831–1916)
Dedekind-Artin theorem
degree of a polynomial
degree over F
DeMoivre's theorem
DeMoivre, A. (1667–-1754)
density theorem
derivative, formal
derived
series
subgroup
higher
Descartes, R. (1596–1650)
Dickson, L. E. (1874–1954)
dicyclic group
difference of two sets
dihedral group
dimension of a vector space
Dirac, P.A.M. (1902–1984)
direct product
direct sum
characterization
external
internal
Dirichlet, P.G.L. (1805–1859)
discriminant
of a cubic
of a polynomial
of a quadratic
disjoint sets
divisible group
division algorithm
for integers
for polynomials
division ring
divisor
common
greatest common
domain
Ore
domain of a mapping
Doyle, A.C. (1859–1930)
duplicating a cube
Eisenstein criterion
element of a set
elementary divisors
elementary symmetric polynomial
elements of sets
embedding theorem
empty set
endomorphism
endomorphism ring
equivalence
afforded by a partition
class
kernel of
logical
natural mapping
quotient set of
relation
Euclid of Alexandria (ca 330-275 BC)
euclidean algorithm
euclidean domain
Euler function
Euler, L. (1707–1783)
evaluation map
exponent laws
extension of fields
abelian
algebraic
closures in
cyclic
dimension of
F-automorphism
finite
Galois
Galois group
intermediate field
main theorem of Galois theory
norm
normal
primitive element theorem
radical
separable
separable closure
simple
trace
factor group
factor modules
factor ring
ideals in
factor theorem
factorial
factorization
trivial
FC-group
Feit, W. (1930–2004)
Fermat's theorem
Fermat, P. (1601–1665)
Ferrari, L. (1522–1565)
Ferro, S. (1522–1565)
field
algebraic closure
algebraic element
algebraically closed
Dedekind's lemma
Dedekind-Artin theorem
extension
finite
Galois
minimal polynomial of an element
of algebraic numbers
of constructible numbers
of quotients
perfect
prime
splitting
transcendental element
field of quotients
figure
motion of
finite field
cyclic unit group
existence
primitive elements
subfields
uniqueness
finite subgroup test
finitely generated group
module
Fitting's lemma
Fitting, H. (1906–1938)
Fitting;s theorem
fixed field of a group
four square identity
Fourier, J.B.J. (1768–1830)
Frattini argument
Frattini subgroup
Frattini, G. (1852–1925)
free module
Frobenius
automorphism
endomorphism
theorem
Frobenius, G. (1849–1917)
fully invariant submodule
fundamental identities of a mapping
fundamental theorem
finitely generated modules over a PID
for finite abelian groups
for vector spaces
of algebra
of cyclic groups
of Galois theory
of symmetric polynomials
Gödel, K. (1906–1978)
Galois connection
Galois extension
Galois field
Galois group of a polynomial
Galois group of an extension
Galois theory
main theorem
Galois' criterion
Galois, E. (1811–1832)
Gauss, C.F. (1777–1855)
formula
Gaussian integers
gcd
integers
polynomials
general associativity
general linear group
generator of a cyclic group
generators
of a field extension
of a group
of a subgroup
Goldbach, C. (1690–1764)
Goldie, A.W. (1920–2005)
Grassmann, H.G. (1809–1877)
greatest common divisor
group
abelian
actions (G-sets)
alternating
automorphism
Burnside-Wielandt theorem
Cauchy's theorem
Cauchy-Frobeinius Lemma
Cayley's theorem
center of
central series
character
circle group
class equation
composition series, length, factors
core of a subgroup
correspondence theorem
cosets of a subgroup
cyclic
dicyclic
dihedral
direct product
divisible
extended Cayley theorem
extension problem
factor group of
FC-group
finite abelian p-groups
finite p-groups
finitely generated
Frattini subgroup
G-set
general linear
Hall's theorem
holomorph
homomorphism
image
isomorphic
isomorphism
isomorphism theorem
Jordan–Hölder theorem
Klein
lattice diagram
maximal normal subgroup
metabelian
metacyclic
nilpotent
nongenerators of
normalizer
octic
of motions
of units
p-group
permutation
polycyclic
Prüfer
product of subgroups
product of subsets
projective special linear
quaternion
relations in
Schur's theorem
Schur-Zassenhaus theorem
second isomorphism theorem
semidirect product
simple
simple factor
solvable
special linear
subgroup of
Sylow theorems
symmetric
third isomorphism theorem
torsion
torsion-free
translations of
group action
by conjugation
by multiplication
faithful
fixed element
fixed subset
fixer of
G-morphisms
orbit decomposition theorem
orbit in
stabilizer
transitive
trivial
group of units
Hölder, O. (1859–1937)
Hall's theorem
Hall, P. (1904–1982)
Hall, P. (1904–1982)
Hamilton, W.R. (1805–1865)
Hamming, R. (1915–1998)
(7,4)-code
bound
code
distance
weight
Hankel, H. (1839–1873)
Hardy, G.H. (1877–1947)
Hermite, C. (1822–1901)
Hertz, H.R. (1857–1894)
higher derived subgroups
Hilbert, D. (1862–1943)
Hobbes, T. (1588–1679)
homomorphism
fixed element of
general ring
group
image of
kernel of
module
preimage of
ring
trivial
Hopkins-Levitzky theorem
hypothesis
ideal
annihilator
left
left/right
maximal
maximal left
prime
principal
proper
zero
idempotent
lifting
identity
homomorphism
mapping
permutation
image of a homomorphism
image of a mapping
implication
inclusion mapping
indeterminant
index of a subgroup
induction
definition by
hypothesis
mathematical
principal of
strong
inductive definition
inductive set
inner automorphism
integers
prime
relatively prime
integers modulo n
integral domain
ACCP
as factor ring
associates in
Bézout
embedding theorem
euclidean
field of quotients
greatest common divisor (gcd)
irrreducible in
least common multiple (lcm)
ordered
positive elements
prime element of
prime ideal of
principal ideal domain (PID)
reducible in
unique factorization domain (UFD)
well-ordered
intermediate field of an extension
internal direct sum
intersection of sets
invariant basis number (IBN)
invariant factors
inverse
in a monoid
left/right
of a complex number
permutation
inverse of a mapping
invertibility theorem
ireducible
irreducible
polynomial
isometry
isomorphic groups
isomorphic rings
isomorphism
of rings
isomorphism theorem
group
module
ring
second group
second ring
third group
third ring
Jacobson radical
Jacobson's theorem
Jacobson, N. (1910–1999)
Jordan, C. (1832–1922)
Jordan, C. (1838–1922)
Jordan–Hölder theorem
group
Jordan-Hölder theorem
module
Kaplansky, I. (1917–2006)
kernel of a homomorphism
Klein group
Klein, F. (1849–1925)
Kronecker delta
Kronecker's theorem
Kronecker, L. (1823–1891)
Kummer, E.E. (1810–1893)
Lagrange
four square identity
interpolation
polynomials
Lagrange's theorem
Lagrange, J.L. (1736–1813)
Laplace, P.S. de (1749–1827)
lattice diagram
lcm
leading coefficient
least common multiple
left ideal
Legendre, A.-M. (1752–1833)
Lindemann, F. (1852–1939)
linear combination
trivial
linear polynomial
Lobachevski, N.I. (1793–1865)
localization
logically equivalent
Möbius function
Möbius inversion formula
main theorem of Galois theory
Mal'cev, A.I. (1909–1967)
mapping
action of
bijective
composite
constant
domain, codomain
fundamental identities
identity
image
image of
inverse of
natural
one-to-one
onto
structure preserving
surjective
well-defined
matrix
identity
m by n
main diagonal
operations
parity check
product
ring
similarity
square
zero
matrix rings
ideals in
matrix units
Maurolico, Francesco (1494–1575)
maximal
subgroup
maximal ideal
maximal normal subgroup
McKay, J.H.
metacyclic group
minimal polynomial over F
modular irreducibility test
modular law
for subgroups
module
annihilator in
artinian
basis of
complemented
composition series, length, factors
direct sum, external
direct sum, internal
endomorphism ring
factor modules
finite dimensional
finitely generated
free
generating set
homomorphism
indecomposable
independent set
independent subset
invariant basis number (IBN)
isomorphic
isomorphism
isomorphism theorem
left
morphism
noetherian
over a PID
principal
projective
rank
rank theorem
semisimple
simple
submodule
sum
torsion-free
trivial morphism
modules over a PID
annihilators in
decomposition of p-modules
free if and only if torsion-free
fundamental theorem
order of elements
p-modules
p-primary component
primary decomposition theorem
submodule theorem
torsion submodule
modulus
modulus of a complex number
monic polynomial
monoid
commutative
monomial
morphism
identity
imagel of
kernel of
module
motion of a figure
multiplication rule for complex numbers
multiplication theorem
multiplicative notation
natural mapping
natural numbers
negative
Newton identities
Newton, I. (1642–1727)
nil radical
nilpotency class
nilpotent
ideal
nilpotent group
Burnside–Wielandt theorem
Noether, E. (1882–1935)
noetherian rings
left
nongenerator in a group
norm in quadratic integers
normal closure
normal subgroup
test for
normalizer
normalizer of a subgroup
number
of elements in a set
octic group
one-to-one mapping
onto mapping
orbit
orbit decomposition theorem
order
of a group
order of an element
finite
in a module over a PID
infinite
ordered integral domain
ordered n-tuple
component of
ordered pair
Ore domains
Ore, O. (1899–1968)
Oresme, Nicole (1323–1382)
orthogonality theorem
p-group
finite
finite abelian
p-module
direct sum decomposition
elementary divisors
type
uniqueness of decomposition
parity
parity check matrix
parity theorem
partial fraction expansion
partial order
partially ordered set (poset)
inductive
upper bound in
partition
cells of
singleton
theorem
trivial
Pascal, B. (1623–1662)
identity
triangle
Peano axioms
Peano, Giuseppe. (1858–1932)
Pell's equation
permutation
cycle
cycle index
disjoint
even
fixed/moved element of
identity
inverse
odd
of a set
sign of
permutation group
PID
and UFD's
module decomposition
module is free iff torsion-free
primes in
PID (principal ideal domain)
pigeonhole principle
Poincaré, H. (1854–1912)
pointwise operations
Poisson, S.-D.(1781–1840)
polycyclic group
polynomial
alternating
coefficients
constant
constant coefficient
cubic
cyclotomic
degree
derivative of
Eisenstein criterion
elementary symmetric
equality
error-locator
evaluation theorem
even/odd
factor is a field
factor rings
factor theorem
fundamental theorem of symmetric polynomials
gcd
homogeneous
homogeneous components
irreducible
Kronecker's theorem
Lagrange
leading coefficient
least common multiple
lexicographic order
linear
minimal
modular irreducibility test
monic
negative of
Newton identities
partial fraction expansion
primitive
principle ideal domain
proper factorization
quadratic
quartic
quintic
rational forms
rational roots theorem
reducible
relatively prime
remainder theorem
repeated root
ring
roots of
separable
several variables
solvable
splits
symmetric
symmetric rational forms
unique factorization theorem
zero
positive elements
positive numbers
power of an element
Prüfer group
preimage
primary component
primary decomposition theorem
for finite abelian groups
for modules over a PID
prime
in a PID
prime factorization
theorem
prime fields
prime ideal
prime number
prime power
prime ring
primitive element
primitive element theorem
primitive polynomial
primitive root modulo p
primitive roots of unity
principal ideal
projection
projections
projective module
projective special linear group
proof
by contradiction
direct method
reduction to cases
proper
ideal
subgroup
submodule
quadratic
integers
quadratic formula
quadratic polynomial
quartic polynomial
quaternion group
quaternions
conjugate
norm
quintic polynomial
quotient
quotient set
radian measure of an angle
radical extension
rank of a module
rank theorem for modules
rational expression
rational forms
rational numbers
rational roots theorem
real numbers
recursion theorem
reducible
in an integral domain
reducible polynomial
regular n-gon
regular representation
relation
relatively prime
integers
polynomials
remainder
remainder theorem
residue class
residue modulo n
ring
automorphism
binomial theorem
boolean
center
characteristic of
Chinese remainder theorem
commutative
corner ring
decomposition theorem
density theorem
direct product
division
endomorphism
factor
general
group of units
homomorphism
ideal of
idempotent in
isomorphism
isomorphism theorem
Jacobson radical
left artinian
left noetherian
lifting idempotents
local
maximal left ideal
negative in
nil radical of
nilpotent ideal
nilpotent in
noetherian
of functions
of matrices
opposite
polynomial
prime
prime ideal of
semiperfect
semisimple
simple
subring
subtraction
unit
unity of
upper triangular
zero ring
root
multiplicity
of a polynomial
of unity
rational
repeated
Ruffini, P. (1765–1822)
Russell, B. (1872–1970)
Schur's lemma
Schur's theorem
Schur, I. (1875–1941)
Schur-Zassenhaus theorem
second isomorphism theorem
for groups
for rings
semidirect product
semiperfect ring
semisimple module
homogeneous
homogeneous component
semisimple rings
separable closure
separable extension
sequence
recursively defined
set
cartesian product
containment
difference
disjoint
element of
empty
equality
infinite
intersection
operations on
proiper containment
singleton
subset of
union
Shannon, C.E. (1916–2001)
sign of a permutation
similar matrics
simple
group
module
ring
solvable group, series
solvable polynomial
span of vectors
special linear group
projective
Split mapping
splitting field
existence
uniqueness
square free
stabilizer
Steinitz exchange lemma
subfield
subgroup
characteristic
conjugate
cyclic
derived
Frattini
generated by a set
generators
index of
maximal
maximal normal
normal
proper
self-conjugate
subnormal
test
torsion
trivial
unconnected set of
subgroup test
submodule
direct sum
fully invariant
maximal
principal
proper
sum of
submodule theorem
subset
subspace test
surjective mapping
Sylow p-subgroup
number of
Sylow theorems
Sylow's first theorem
Sylow's second theorem
Sylow's third theorem
Sylow, L. (1832–1918)
symmetric group
symmetric polynomial
elementary
symmetric rational forms
symmetry of a figure
syndrome
Tartaglia, N. (1500–1557)
theorems
third isomorphism theorem
for groups
for rings
Thompson, J.G. (1932–)
torsion group
torsion subgroup
torsion submodule
torsion-free
element
module
torsion-free group
transcendental element
transposition
trisecting an angle
trivial factorization
trivial homomorphism
trivial linear combination
trivial subgroup
Tucker, A.
UFD
characterization
Gauss' lemma
polynomials
unconnected subgroups
union of sets
unique factorization theorem
unit
circle
in a monoid
properties
unit circle
unity
left/right
unity for a binary operation
upper triangular
variety of groups
vector space
basis of
dependence in
dimension
dimension theorem
finite dimensional
fundamental theorem
invariance theorem
linear combination
linear independence
linear transformation
scalar multiples
spanning set
subspace
zero space
Venn diagrams
Venn, J. (1834–1883)
Voltaire–F.M.A. (1694–1778)
Wedderburn's theorem
on division rings
on simple rings
Wedderburn, J.H.M. (1882–1948)
Wedderburn-Artin Theorem
Weierstrass, K. (1815–1897)
Weil, A. (1906–1998)
well-defined mapping
well-ordered integral domain
well-ordering
axiom
Weyl, H. (1885–1955)
Whitehead, A.N. (1861–1947)
Wielandt, H. (1910–2001)
Wiles, A.
Wilson's theorem
Witt, E., (1911–1991)
word
empty
justaposition
length
Zassenhaus, H. (1912–1991)
Zermelo, E. (1871–1953)
zero
ideal
matrix
vector space
Zorn's lemma
Zorn, M. (1906–1993)