Index Index
Abel, Niels Henrik, Paragraph
Abelian group, Paragraph
Ackermann’s function, Exercise
Adleman, L., Paragraph
Algebraic closure, Paragraph
Algebraic extension, Paragraph
Algebraic number, Paragraph
Algorithm
division, Theorem
Euclidean, Paragraph
Ascending chain condition, Paragraph
Associate elements, Paragraph
Atom, Paragraph
Automorphism
inner, Exercise
Basis of a lattice, Paragraph
Bieberbach, L., Paragraph
Binary operation, Paragraph
Binary symmetric channel, Paragraph
Boole, George, Paragraph
Boolean algebra
atom in a, Paragraph
definition of, Paragraph
finite, Paragraph
isomorphism, Paragraph
Burnside’s Counting Theorem, Theorem
Cancellation law
for groups, Paragraph
for integral domains, Proposition
Cardano, Gerolamo, Paragraph
Carmichael numbers, Exercise
Cauchy, Augustin-Louis, Paragraph
Cauchy’s Theorem, Theorem
Cayley table, Paragraph
Cayley, Arthur, Paragraph
Cayley’s Theorem, Theorem
Centralizer
of a subgroup, Paragraph
Characteristic of a ring, Paragraph
Chinese Remainder Theorem
for integers, Theorem
Cipher, Paragraph
Ciphertext, Paragraph
Circuit
parallel, Paragraph
series, Paragraph
series-parallel, Paragraph
Class equation, Paragraph
Code
BCH, Paragraph
cyclic, Paragraph
group, Paragraph
linear, Paragraph
minimum distance of, Paragraph
polynomial, Paragraph
Commutative diagrams, Paragraph
Commutative rings, Paragraph
Composite integer, Paragraph
Composition series, Paragraph
Congruence modulo \(n\), Example
Conjugacy classes, Paragraph
Conjugate elements, Paragraph
Conjugate, complex, Paragraph
Conjugation, Example
Constructible number, Paragraph
Correspondence Theorem
for groups, Theorem
for rings, Theorem
Coset
leader, Paragraph
left, Paragraph
representative, Paragraph
right, Paragraph
Coset decoding, Paragraph
Cryptanalysis, Paragraph
Cryptosystem
affine, Paragraph
definition of, Paragraph
monoalphabetic, Paragraph
polyalphabetic, Paragraph
private key, Paragraph
public key, Paragraph
RSA, Paragraph
single key, Paragraph
Cycle
definition of, Paragraph
disjoint, Paragraph
De Morgan, Augustus, Paragraph
De Morgan’s laws
for Boolean algebras, Item
for sets, Theorem
Decoding table, Paragraph
Deligne, Pierre, Paragraph
DeMoivre’s Theorem, Theorem
Derivative, Paragraph
Determinant, Vandermonde, Paragraph
Dickson, L. E., Paragraph
Diffie, W., Paragraph
Direct product of groups
external, Paragraph
internal, Paragraph
Discriminant
of the cubic equation, Exercise
of the quadratic equation, Exercise
Division algorithm
for integers, Theorem
for polynomials, Theorem
Division ring, Paragraph
Domain
Euclidean, Paragraph
principal ideal, Paragraph
unique factorization, Paragraph
Doubling the cube, Paragraphs
Eisenstein’s Criterion, Theorem
Element
associate, Paragraph
identity, Item
inverse, Item
irreducible, Paragraph
order of, Paragraph
prime, Paragraph
primitive, Paragraph
transcendental, Paragraph
Equivalence class, Paragraph
Equivalence relation, Paragraph
Euclidean algorithm, Paragraph
Euclidean domain, Paragraph
Euclidean group, Paragraph
Euclidean inner product, Paragraph
Euclidean valuation, Paragraph
Euler \(\phi\)-function, Paragraph
Extension
algebraic, Paragraph
field, Paragraph
finite, Paragraph
normal, Paragraph
radical, Paragraph
simple, Paragraph
External direct product, Paragraph
Faltings, Gerd, Paragraph
Fermat’s factorization algorithm, Exercise
Fermat’s Little Theorem, Theorem
Ferrari, Ludovico, Paragraph
Ferro, Scipione del, Paragraph
Field, Paragraph
algebraically closed, Paragraph
base, Paragraph
extension, Paragraph
fixed, Paragraph
Galois, Paragraph
of fractions, Paragraph
of quotients, Paragraph
splitting, Paragraph
Finitely generated group, Paragraph
Fior, Antonio, Paragraph
First Isomorphism Theorem
for groups, Theorem
for rings, Theorem
Fixed point set, Paragraph
Freshman’s Dream, Lemma
Function
bijective, Paragraph
composition of, Paragraph
definition of, Paragraph
domain of, Paragraph
identity, Paragraph
injective, Paragraph
invertible, Paragraph
one-to-one, Paragraph
onto, Paragraph
range of, Paragraph
surjective, Paragraph
Fundamental Theorem
of Arithmetic, Theorem
of Finite Abelian Groups, Theorem
Fundamental Theorem of Galois Theory, Theorem
Galois field, Paragraph
Galois group, Paragraph
Gauss, Karl Friedrich, Paragraph
Gaussian integers, Example
Gauss’s Lemma, Theorem
Generator of a cyclic subgroup, Paragraph
Generators for a group, Paragraph
Glide reflection, Paragraph
Gorenstein, Daniel, Paragraph
Greatest common divisor
of two integers, Paragraph
of two polynomials, Paragraph
Greatest lower bound, Paragraph
Greiss, R., Paragraph
Grothendieck, Alexander, Paragraph
Group
abelian, Paragraph
action, Paragraph
alternating, Paragraph
center of, Paragraph
circle, Paragraph
commutative, Paragraph
cyclic, Paragraph
definition of, Paragraph
dihedral, Paragraph
Euclidean, Paragraph
factor, Paragraph
finite, Paragraph
finitely generated, Paragraph
Galois, Paragraph
generators of, Paragraph
homomorphism of, Paragraph
infinite, Paragraph
isomorphic, Paragraph
isomorphism of, Paragraph
nonabelian, Paragraph
noncommutative, Paragraph
of units, Example
order of, Paragraph
orthogonal, Paragraph
permutation, Paragraph
point, Paragraph
quaternion, Example
quotient, Paragraph
solvable, Paragraph
space, Paragraph
special orthogonal, Paragraph
symmetric, Paragraph
symmetry, Paragraph
Gödel, Kurt, Paragraph
Hamming distance, Paragraph
Hamming, R., Paragraph
Hellman, M., Paragraph
Homomorphic image, Paragraph
Homomorphism
kernel of a group, Paragraph
kernel of a ring, Paragraph
of groups, Paragraph
ring, Paragraph
Ideal
definition of, Paragraph
maximal, Paragraph
one-sided, Remark
prime, Paragraph
principal, Paragraph
trivial, Example
two-sided, Remark
Indeterminate, Paragraph
Index of a subgroup, Paragraph
Induction
first principle of, Principle
second principle of, Principle
Infimum, Paragraph
Inner product, Paragraph
Integral domain, Paragraph
Internal direct product, Paragraph
International standard book number, Exercise
Irreducible element, Paragraph
Irreducible polynomial, Paragraph
Isometry, Paragraph
Isomorphism
of Boolean algebras, Paragraph
of groups, Paragraph
ring, Paragraph
Kernel
of a group homomorphism, Paragraph
of a ring homomorphism, Paragraph
Key
definition of, Paragraph
private, Paragraph
public, Paragraph
single, Paragraph
Kronecker, Leopold, Paragraph
Kummer, Ernst, Paragraph
Lagrange’s Theorem, Theorem
Laplace, Pierre-Simon, Paragraph
Lattice
completed, Paragraph
definition of, Paragraph
distributive, Paragraph
Lattice of points, Paragraph
Lattices, Principle of Duality for, Axiom
Least upper bound, Paragraph
Left regular representation, Paragraph
Linear combination, Paragraph
Linear dependence, Paragraph
Linear independence, Paragraph
Linear map, Paragraph
Linear transformation
Lower bound, Paragraph
Mapping. See Function
Matrix
distance-preserving, Paragraph
generator, Paragraph
inner product-preserving, Paragraph
invertible, Paragraph
length-preserving, Paragraph
nonsingular, Paragraph
null space of, Paragraph
orthogonal, Paragraph
parity-check, Paragraph
similar, Example
unimodular, Paragraph
Matrix, Vandermonde, Paragraph
Maximal ideal, Paragraph
Maximum-likelihood decoding, Paragraph
Meet, Paragraph
Minimal generator polynomial, Paragraph
Minimal polynomial, Paragraph
Minkowski, Hermann, Paragraph
Monic polynomial, Paragraph
Mordell conjecture, Paragraph
Multiplicity of a root, Paragraph
Noether, A. Emmy, Paragraph
Noether, Max, Paragraph
Normal extension, Paragraph
Normal series of a group, Paragraph
Normal subgroup, Paragraph
Normalizer, Paragraph
Null space
of a matrix, Paragraph
Odd Order Theorem, Theorem
Orbit, Paragraph
Orthogonal group, Paragraph
Orthogonal matrix, Paragraph
Orthonormal set, Paragraph
Partial order, Paragraph
Partially ordered set, Paragraph
Partitions, Paragraph
Permutation
cycle structure of, Exercise
even, Paragraph
odd, Paragraph
Permutation group, Paragraph
Plaintext, Paragraph
Polynomial
code, Paragraph
content of, Paragraph
definition of, Paragraph
degree of, Paragraph
error, Paragraph
error-locator, Exercise
greatest common divisor of, Paragraph
irreducible, Paragraph
leading coefficient of, Paragraph
minimal, Paragraph
minimal generator, Paragraph
monic, Paragraph
primitive, Paragraph
root of, Paragraph
separable, Paragraph
zero of, Paragraph
Polynomial in \(n\) indeterminates, Paragraph
Polynomial separable, Paragraph
Poset
definition of, Paragraph
largest element in, Paragraph
smallest element in, Paragraph
Power set, Example
Prime element, Paragraph
Prime ideal, Paragraph
Prime integer, Paragraph
Primitive element, Paragraph
Primitive Element Theorem, Theorem
Primitive polynomial, Paragraph
Principal ideal, Paragraph
Principal ideal domain (PID), Paragraph
Principal series, Paragraph
Pseudoprime, Exercise
Resolvent cubic equation, Exercise
Ring
characteristic of, Paragraph
commutative, Paragraph
definition of, Paragraph
division, Paragraph
factor, Paragraph
homomorphism, Paragraph
isomorphism, Paragraph
Noetherian, Paragraph
quotient, Paragraph
with identity, Paragraph
with unity, Paragraph
Rivest, R., Paragraph
RSA cryptosystem, Paragraph
Ruffini, P., Paragraph
Russell, Bertrand, Paragraph
Scalar product, Paragraph
Second Isomorphism Theorem
for groups, Theorem
for rings, Theorem
Shamir, A., Paragraph
Shannon, C., Paragraph
Sieve of Eratosthenes, Exercise
Simple extension, Paragraph
Simple group, Paragraph
Simple root, Paragraph
Solvability by radicals, Paragraph
Spanning set, Paragraph
Splitting field, Paragraph
Squaring the circle is impossible, Paragraphs
Standard decoding, Paragraph
Subgroup
centralizer, Paragraph
commutator, Paragraph
cyclic, Paragraph
definition of, Paragraph
index of, Paragraph
isotropy, Paragraph
normal, Paragraph
normalizer of, Paragraph
proper, Paragraph
stabilizer, Paragraph
Sylow\(p\)-subgroup, Paragraph
translation, Paragraph
trivial, Paragraph
Subgroup!\(p\)-subgroup, Paragraph
Subnormal series of a group, Paragraph
Subring, Paragraph
Supremum, Paragraph
Switch
closed, Paragraph
definition of, Paragraph
open, Paragraph
Sylow \(p\)-subgroup, Paragraph
Sylow, Ludvig, Paragraph
Tartaglia, Paragraph
Third Isomorphism Theorem
for groups, Theorem
for rings, Theorem
Transcendental element, Paragraph
Transcendental number, Paragraph
Transposition, Paragraph
Trisection of an angle, Subsection
Unique factorization domain (UFD), Paragraph
Universal Product Code, Exercise
Upper bound, Paragraph
Vandermonde determinant, Paragraph
Vandermonde matrix, Paragraph
Vector space
basis of, Paragraph
definition of, Paragraph
dimension of, Paragraph
subspace of, Paragraph
