Index Index
Abelian group, Paragraph
Ackermann's function, Exercise
Adleman, Leonard, Paragraph
Algorithm
division, Theorem
Euclidean, Paragraph
Ascending chain condition, Paragraph
Associate elements, Paragraph
Cancellation law
for groups, Paragraph
for integral domains, Proposition
Cardano, Gerolamo, Paragraph
Carmichael numbers, Exercise
Cauchy, Augustin-Louis, Paragraph
Cayley table, Paragraph
Centralizer
of a subgroup, Paragraph
Characteristic of a ring, Paragraph
Chinese Remainder Theorem
for integers, Theorem
Cipher, Paragraph
Ciphertext, Paragraph
Class equation, Paragraph
Commutative rings, Paragraph
Composite integer, Paragraph
Congruence modulo \(n\), Example
Conjugacy classes, Paragraph
Conjugate, complex, Paragraph
Conjugation, Example
Correspondence Theorem
for rings, Theorem
Coset
left, Paragraph
representative, Paragraph
right, 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's laws
for sets, Theorem
DeMoivre's Theorem, Theorem
Diffie, Whitfield, 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
Eisenstein's Criterion, Theorem
Element
associate, Paragraph
identity, Item
inverse, Item
irreducible, Paragraph
order of, Paragraph
prime, Paragraph
Equivalence class, Paragraph
Equivalence relation, Paragraph
Euclidean algorithm, Paragraph
Euclidean domain, Paragraph
Euclidean valuation, Paragraph
Euler \(\phi\)-function, Paragraph
Euler, Leonhard, Paragraph
Feit, Walter, Paragraph
Fermat's factorization algorithm, Exercise
Fermat, Pierre de, Paragraph
Ferrari, Ludovico, Paragraph
Ferro, Scipione del, Paragraph
Field, Paragraph
of fractions, Paragraph
of quotients, Paragraph
Fior, Antonio, Paragraph
First Isomorphism Theorem
for rings, Theorem
Fixed point set, Paragraph
forward secrecy, Paragraph
Function
bijective, Paragraph
Boolean, 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
switching, Paragraph
Fundamental Theorem
of Arithmetic, Theorem
Galois, Évariste, Paragraph
Gauss's Lemma, Theorem
Gauss, Karl Friedrich, Paragraph
Gaussian integers, Example
Generator of a cyclic subgroup, Paragraph
Greatest common divisor
in an integral domain, Paragraph
of two integers, Paragraph
of two polynomials, Paragraph
Group
abelian, Paragraph
action, Paragraph
alternating, Paragraph
center of, Paragraph
circle, Paragraph
commutative, Paragraph
cyclic, Paragraph
definition of, Paragraph
dihedral, Paragraph
finite, Paragraph
general linear, Example
infinite, Paragraph
nonabelian, Paragraph
noncommutative, Paragraph
of units, Example
order of, Paragraph
permutation, Paragraph
quaternion, Example
special linear, Example
symmetric, Paragraph
Hellman, Martin, Paragraph
Hilbert, David, Paragraph
Homomorphism
canonical, Paragraph
evaluation, Example
kernel of a ring, Paragraph
natural, 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
Integral domain, Paragraph
International Standard Book Number (ISBN), Paragraph
Irreducible
distinguished, Paragraph
Irreducible element, Paragraph
Irreducible polynomial, Paragraph
Isomorphism
ring, Paragraph
Kernel
of a ring homomorphism, Paragraph
Key
definition of, Paragraph
private, Paragraph
public, Paragraph
single, Paragraph
Lagrange's Theorem, Theorem
Laplace, Pierre-Simon, Paragraph
Lie, Sophus, Paragraph
Linear transformation
definition of, Example
Orbit, Paragraph
Partitions, Paragraph
Permutation
even, Paragraph
odd, Paragraph
Permutation group, Paragraph
Plaintext, Paragraph
Polynomial
content of, Paragraph
definition of, Paragraph
degree of, Paragraph
greatest common divisor of, Paragraph
irreducible, Paragraph
leading coefficient of, Paragraph
monic, Paragraph
primitive, Paragraph
root of, Paragraph
zero of, Paragraph
Polynomial!in \(n\) indeterminates, Paragraph
Prime element, Paragraph
Prime ideal, Paragraph
Prime integer, Paragraph
Primitive \(n\)th root of unity, Paragraph
Primitive polynomial, Paragraph
Principal ideal, Paragraph
Principal ideal domain (PID), Paragraph
Pseudoprime, Exercise
Resolvent cubic equation, Exercise
Rigid motion, Paragraph
Ring
characteristic of, Paragraph
commutative, Paragraph
definition of, Paragraph
division, Paragraph
factor, Paragraph
homomorphism, Paragraph
isomorphism, Paragraph
Noetherian, Paragraph
quotient, Paragraph
Rivest, Ron, Paragraph
RSA cryptosystem, Paragraph
Second Isomorphism Theorem
for rings, Theorem
Shamir, Adi, Paragraph
Sieve of Eratosthenes, Exercise
Subgroup
centralizer, Paragraph
cyclic, Paragraph
definition of, Paragraph
index of, Paragraph
isotropy, Paragraph
proper, Paragraph
stabilizer, Paragraph
trivial, Paragraph
Subring, Paragraph
Switching function, Paragraph
Tartaglia, Paragraph
Third Isomorphism Theorem
for rings, Theorem
Thompson, John, Paragraph
Transposition, Paragraph