Index Index
\((G,*)\), Definition
\(*\), Definition
\(<\) (groups), Definition
\([G:H]\), Definition
\(\Aut(G)\), Definition
\(\Aut(K)\), Definition
\(\Aut(K/F)\), Definition
\(\C\), Item
\(\cap\), Definition
\(\cup\), Definition
\(\E\), Item
\(\End(_{Ab}A)\), Item
\(\End_R(M)\), Definition
\(\Fun(X,Y)\), Remark
\(\gcd\) (in a PID), Definition
\(\gcd\) (of a matarix), Definition
\(\GL\), Item
\(\Hom_R\), Definition
\(\Hom_{AB}(G,H)\), Exercise
\(\Inn(G)\), Definition
\(\ker(\varphi)\) (groups), Definition
\(\langle S \rangle\), Definition
\(\langle x \rangle\), Item
\(\leq\) (groups), Definition
\(\N\), Item
\(\nsg\), Definition
\(\O\), Item
\(\operatorname{Fun}(X,R)\), Item
\(\Orb_G(s)\), Definition
\(\Perm(S)\), Definition
\(\Q(\sqrt d)\), Lemma
\(\Q\), Item
\(\R\), Item
\(\rtimes\), Definition
\(\sigma\), Definition
\(\SL\), Definition
\(\sse\), Item
\(\Stab_G(s)\), Definition
\(\subset\), Item
\(\Syl_p(G)\), Definition
\(\Z\), Item
\(\{e\}\), Item
\(A_n\), Definition
\(C_\infty\), Convention
\(C_G(x)\), Definition
\(C_n\), Convention
\(D_{2n}\), Definition
\(e\), Item
\(F(\a)\), Definition
\(F[a]\), Definition
\(G'\), Example
\(G\)-action, Definition
\(G^\times\), Example
\(G^{op}\), Item
\(HK\), Definition
\(IJ\), Item
\(L^G\), Definition
\(n\th\) power, Definition
\(n\Z\), Item
\(N_G(S)\), Definition
\(n_p\), Definition
\(p\)-group, Definition
\(R[G]\), Definition
\(R\)-algebra, Definition
\(R\)-linear combination, Definition
\(R\)-map, Definition
\(R\)-module, Definition
\(R\)-module homomorphism, Definition
\(R\)-module isomorphism, Definition
\(R^{op}\), Item
\(S\inv R\), Definition
\(S_n\), Definition
\(V_g\), Definition
\(Z(G)\), Definition
\(Z_G(g)\), Definition
\(|G|\), Item
\(|x|\), Item
abelian group, Definition
absorption, Item
action (of a module), Definition
adjoining an element, Definition
algebra, Definition
algebraic closure (of a field), Definition
algebraic element, Definition
algebraic field extension, Definition
algebraically closed field, Definition
alternating group, Definition
Artin’s theorem, Theorem
ascending chain condition (ACC), Definition
associate, Definition
associative property, Item
automorphism, Definition
automorphism group, Definition
automorphism group (of a field extension), Definition
automorphism group (of a field), Definition
basis, Definition
bijective, Item
binary operation, Definition
binomial coefficient, Definition
binomial theorem, Theorem
binomial theorem (for commutative rings), Theorem
block diagonal matrix, Definition
cancellation (groups), Item
cancellation (in integral domains), Lemma
cardinality, Definition
Cartesian product, Definition
Cartesian product (of rings), Item
Cauchy’s theorem, Theorem
Cayley-Hamilton theorem, Theorem
Cayley’s theorem, Theorem
center (of a group), Definition
center (of a ring), Definition
centralizer, Definition
change of basis matrix, Definition
characteristic (of a ring), Definition
characteristic map, Item
Chinese Remainder Theorem, Remark
class equation, Theorem
classification of cyclic groups, Theorem
classification of finitely generated modules over a PID (elementary divisor form), Theorem
classification of finitely generated modules over a PID (invariant factor form), Theorem
classification of finitely generated vector spaces, Theorem
classification of groups of order \(pq\), Theorem
cokernel (of \(R\)-module homomorphism, Definition
commutative property, Definition
commutative ring, Definition
commutator subgroup, Example
complex numbers, Item
composition of functions, Definition
conjugacy class, Definition
conjugate, Definition
conjugation action, Item
constant map, Item
content, Definition
coset, Definition
countably infinite set, Definition
counting elements, Example
criteria for separability, Theorem
cycle, Definition
cycle decompostion, Item
cyclic group, Definition
cyclic module, Definition
cyclotomic polynomial, Proposition
De Morgan’s laws (sets), Theorem
degree (of a field extension), Definition
degree (of a polynomial), Definition
degree formula, Theorem
derivative (of a polynomial), Definition
determinant map, Item
diagonalizable (linear operator), Definition
dihedral group, Definition
dimension (of a vector space), Definition
dimension theorem, Theorem
direct product (of cyclic groups), Example
direct product (of groups), Definition
direct sum (of groups), Definition
disjoint cycles, Proposition
distributive law (sets), Theorem
distributive property, Item
divides, Definition
division algorithm (for polynomial rings), Theorem
division algorithm (integers), Theorem
division ring, Definition
domain of a function, Definition
Eisenstein’s criterion, Theorem
element counting argument, Example
elementary basis change operation, Definition
elementary column operation, Definition
elementary divisor form (of a group), Definition
elementary divisors (of a group), Definition
elementary divisors (of a module), Definition
elementary matrix, Definition
elementary row operation, Definition
endomorphism (of modules), Definition
endomorphism ring, Item
equality of \(F[x]\)-modules, Proposition
equivalence class, Definition
equivalence relation, Definition
equivalence relation induced by a group action, Definition
Euclidean domain (ED), Definition
evaluation homomorphism, Exercise
even permutation, Definition
exchange lemma, Lemma
exponential map, Item
extension (of fields), Definition
extension of a function, Item
external direct product, Definition
external semidirect product (of groups), Definition
factor theorem, Theorem
faithful action, Definition
family of sets, Definition
Fermat’s little theorem, Exercise
field, Definition
field extension, Definition
field of fractions, Definition
finite dimensional (as a vector space), Definition
finite set, Definition
finitely generated (ideal), Definition
finitely generated group, Definition
finitely generated module, Definition
first isomorphism theorem (for modules), Item
first isomorphism theorem (for rings), Theorem
first isomorphism theorem for groups, Theorem
fixed subfield, Definition
fraction field, Definition
free \(R\)-module, Definition
free module, Definition
free rank (of a module), Definition
function, Definition
fundamental theorem of finitely generated abelian groups (FTFGAG), Theorem
fundamental theorem of Galois theory (FTGT), Theorem
Galois correspondance, Theorem
Galois field extension, Definition
Galois group, Definition
Gaussian integers, Item
general linear group, Item
generalized associative law, Exercise
generated cyclic subgroup, Item
generated ideal, Definition
generated module, Definition
generated normal subgroup, Item
generated subfield, Definition
generated subgroup, Definition
generating set, Definition
generator, Definition
greatest common divisor (gcd) (in a ED), Definition
greatest common divisor (in a PID), Definition
greatest common divisor (of a matrix), Definition
group, Definition
group action, Definition
group action argument, Example
group homomorphism, Definition
group of units, Example Definition
group ring, Definition
homomorphism (of rings), Definition
IASN, Theorem
ideal, Definition
ideal generated by \(A\), Definition
idempotent (ring element), Definition
identity element, Item
identity map, Item
identity property, Item
image, Item
inclusion map, Item
index, Definition
index tower, Theorem
indexed family of sets, Definition
injective, Item
inner automorphism, Definition
inner automorphism group, Definition
integers, Item
integers modulo \(n\), Definition
integral domain, Definition
intermediate field, Definition
internal direct product, Definition
internal semidirect product (of groups), Definition
intersection, Definition
invariant factor form (of a group), Definition
invariant factors (of a group), Definition
invariant factors (of a module), Definition
inverse element, Item
inverse property, Item
invertible function, Item
irreducible (polynomial), Definition
irreducible element (in integral domains), Definition
isometry, Definition
isomorphic, Definition
isomorphism, Definition
isomorphism (of rings), Definition
isomorphism invariant, Theorem
Jordan block, Definition
Jordan canonical form (JCF), Theorem
kernel (of an \(R\)-module homomorphism), Definition
kernel of a group homomorphism, Definition
Lagrange’s theorem, Theorem
Lagrange’s theorem (converse is false), Exercise
lattice isomorphism theorem (for modules), Item
lattice isomorphism theorem (for rings), Theorem
lattice isomorphism theorem for groups, Theorem
left coset, Definition
left module, Definition
left multiplication action, Item
left regular action, Item
linear combination, Definition
linear transformation, Definition
linearly dependent, Definition
linearly independent, Definition
LOIS, Theorem
map, Definition
matrix of a free module homomorphism, Definition
matrix of a linear transformation, Definition
maximal ideal, Definition
maximal linearly independent, Exploration
minimum polynomial (of a field extension), Definition
minimum polynomial (of a linear transformation), Definition
minimum polynomial (of a matrix), Definition
minor (of a matrix), Definition
modular law (of ideals), Exercise
module, Definition
module homomorphism, Definition
module isomorphism, Definition
monic polynomial, Definition
multiple, Definition
multiplicatively closed subset of non zerodivisors, Definition
mutliplicity of a root, Definition
natural log map, Item
natural numbers, Item
nilpotent (ring element), Definition
nilradical, Exploration
noetherian ring, Definition
norm function, Definition
normal subgroup, Definition
normalizer, Definition
normalizer argument, Example
nullity, Definition
nullspace, Definition
odd permutation, Definition
one-to-one, Item
onto, Item
opposite group, Item
opposite ring, Item
orbit, Definition
orbit-stabilizer theorem, Theorem
order of a group, Item
order of a group element, Item
PAN, Theorem
partition, Definition
permutation, Definition
permutation group, Definition
permutation representation, Theorem
polynomial ring, Definition Definition
porism, Corollary
preimage, Item
presentation (of an \(R\)-module), Definition
primary decomposition (of a group), Definition
primative element, Definition
prime element (in integral domains), Definition
prime field, Definition
prime ideal, Definition
prime integer, Definition
principal ideal, Definition
principal ideal domain (PID), Definition
product (groups), Definition
product inclusion map, Definition
projection map, Item
proper ideal, Definition
proper subgroup, Definition
proper subset, Item
quadratic field, Lemma
quaternion group, Item
quotient group, Definition
quotient map, Definition
quotient map (groups), Definition
quotient map (modules), Definition
quotient module, Definition
quotient ring, Definition
range of a function, Definition
rank (of an \(R\)-module), Definition
rank-nullity theorem, Theorem
rational canonical form (RCF), Theorem
rational numbers, Item
real numbers, Item
recognition theorem (for direct products), Theorem
recognition theorem (for internal semidirect products), Theorem
reduction homomorphism, Exercise
reflection, Item
reflexive relation, Item
regular \(n\)-gon, Definition
residue field, Definition
restriction of a function, Item
restriction of scalars, Theorem
right coset, Definition
right module, Definition
ring, Definition
ring arithmetic, Proposition
ring homomorphism, Definition
ring isomorphism, Definition
ring map, Definition
ring of scalars, Paragraph
ring with identity, Definition
ring with unity, Definition
root multiplicity, Definition
rotation, Item
scalars (of a module), Definition
second isomorphism theorem (for modules), Item
second isomorphism theorem (for rings), Theorem
second isomorphism theorem for groups, Theorem
self-conjugation, Item
semidirect product (of groups), Definition
separable field extension, Definition
separable polynomial, Definition
set equality, Item
sign homomorphism, Item
similar \(R\)-module homomorphisms, Definition
similar matrices, Definition
simple field extension, Definition
simple group, Definition
simple module, Exploration
Smith Normal Form (SNF), Theorem
span, Definition
special linear group, Definition
splitting field, Definition
stabilizer, Definition
standard free module, Item
structure homomorphism, Definition
sub-vector space, Paragraph
subfield, Definition
subfield fixed by \(G\), Definition
subfield generated by \(A\), Definition
subgroup, Definition
subgroup test, Theorem
submodule, Definition
submodule generated by \(A\), Definition
subring, Definition
subring test, Lemma
subset, Item
subspace, Paragraph
Sunzi’s remainder theorem (for commutative rings), Theorem
Sunzi’s remainder theorem (for groups), Theorem
surjective, Item
Sylow \(p\)-subgroup, Definition
Sylow subgroup, Definition
Sylow’s theorems, Theorem
symmetric group, Definition
symmetric relation, Item
symmetry, Definition
Taylor Swift, Quotation
third isomorphism theorem (for modules), Item
third isomorphism theorem (for rings), Theorem
third isomorphism theorem for groups, Theorem
torision subgroup, Item
transcendental element, Definition
transitive action, Definition
transitive relation, Item
transposition, Definition
trivial action, Item
trivial group, Item
trivial homomorphism, Item
trivial ring, Item
two-sided ideal, Definition
UMP for a cyclic group, Proposition
UMP for free \(R\)-modules, Theorem
UMP for polynomial rings, Theorem
UMP for quotient groups, Theorem
UMP for quotient modules, Item
UMP for quotient rings, Theorem
uncountably infinite set, Definition
union, Definition
unique factorization domain (UFD), Definition
uniqueness of (module) rank over commutative rings, Theorem
unit, Definition
unital ring, Definition
Universal mapping property for free \(R\)-modules, Theorem
Universal mapping property for quotient modules, Item
universal mapping property for quotient rings, Theorem
universal mapping property of group rings, Theorem
vector space, Definition
well defined function, Definition
