Table of Tableaux
- 1.1, Definition:
divisor, quotient
- 1.2, Theorem 1:
integer division with remainder
- 1.3, Definition:
(greastest) common divisor, relatively prime
- 1.4, Definition:
(least) common multiple
- 2.1, Algorithm:
Euclid's algorithm
- 2.2, Algorithm:
extended Euclidean algorithm
- 2.3, Theorem:
characterization of the gcd
- 2.4, Corollary:
characterization of relative primality
- 2.4, Proposition:
divisibility of a factor
- 3.1, Lemma:
homogeneous diophantine equation, relativele prime case
- 3.1, Theorem:
homogeneous diophantine equation
- 3.2, Algorithm:
linear diophantine equation
- 4.1, Definition:
prime
- 4.1, Theorem:
there are infinitely many primes
- 4.2, Algorithm:
Eratosthenes' sieve
- 4.2, Fact:
Prime number theorem
- 4.3, Theorem:
characterization of primes
- 4.3, Corollary:
a prime divides a factor of a factorization of integers
- 5.1, Theorem:
unique factorization of integers
- 5.2, Theorem:
gcd and lcm in terms of factorizations
- 1.1, Definition:
congruence
- 1.2, Definition:
modular addition and multiplication for integers
- 1.3, Definition:
invertibility
- 1.3, Theorem:
characterization of invertibility
- 2.1, Theorem:
linear congruence
- 2.2, Algorithm:
linear congruence
- 2.3, Theorem:
Chinese remainder theorem
- 3.1, Definition:
a-ary representation of an integer
- 3.1, Theorem:
a-ary representation of an integer
- 4.1, Theorem:
Fermat's little theorem
- 4.2, Definition:
RSA ingredients
- 4.3, Definition:
encoding and decoding in RSA
- 5.1, Definition:
radar detection pattern
- 5.2, Definition:
primitive element
- 5.2, Fact:
existence of primitive elements in prime order fields
- 5.3, Algorithm:
construction of detection patterns
- 1.1, Definition:
polynomial, term, coefficient, monomial
- 1.2, Definition:
sum and product of polynomials
- 2.1, Definition:
degree, leading term, leading coefficient
- 2.2, Definition:
divisor and quotient
- 2.2, Theorem:
quotient and remainder
- 2.3, Definition:
(greatest) common divisor, (least) common multiple
- 2.3, Proposition:
properties of the gcd
- 2.4, Algorithm 1:
Euclid' algorithm
- 2.4, Algorithm 2:
extended Euclidean algorithm
- 2.5, Proposition:
characterization of gcd
- 3.1, Definition:
polynomial function, zero of a polynomial
- 3.2, Theorem:
Lagrange interpolation theorem
- 4.1, Lemma:
zeros of polynomials correspond to linear factors
- 4.1, Definition:
(ir)reducibility
- 4.2, Lemma:
characterization of relative primality
- 4.2, Theorem:
divisibility of a factor
- 4.3, Corollary:
an irreducible polynomial divides a factor of a factorization
- 4.3, Theorem:
unique factorization of polynomials
- 5.1, Definition:
shift register, shift polynomial
- 5.2, Theorem:
polynomial output description of a shift register
- 5.3, Fact:
minimal polynomial of the transition matrix of a shift register
- 5.3, Theorem:
shift register period theorem
- 1.1, Definition:
congruence for polynomials
- 1.1, Theorem:
congruence is an equivalence relation
- 1.2, Definition:
residue classes, residue class ring
- 1.3, Theorem:
unique residue class representatives
- 2.1, Definition:
sum and product in the residue class ring
- 2.2, Theorem 1:
arithmetical rules with 0 and 1
- 2.2, Theorem 2:
general arithmetical rules
- 2.3, Lemma:
The constants in a residue class ring
- 2.4, Theorem:
The residue class ring as a vector space
- 2.5, Theorem:
taking the unique representative is a projection map
- 3.1, Theorem:
n-th order approximations viewed as computations modulo Xn+1
- 3.2, Theorem:
arithmetic modulo an integer in a polynomial ring coincides with
arithmetic modulo that integer in the coefficient ring
- 4.1, Definition:
invertibility and inverse in a residue class ring
- 4.2, Theorem:
characterization of invertibility
- 4.2, Corollary:
residue class rings modulo irreducible polynomials are fields
- 5.1, Theorem:
a finite field has prime power order
- 5.1, Fact:
any finite field is a residue class ring
- 5.2, Theorem:
p-th powers in finite fields of characteristic p
- 5.2, Fact:
A finite residue class field has a primitive element
- 6.1, Definition:
coding theory
- 6.2, Definition:
code, code word, linear code
- 6.2, Definition 2:
distance, minimal distance of a code
- 6.3, Definition:
cyclic code
- 6.4, Theorem:
cyclic decoding theorem
- 6.5, Fact:
BCH bound for cyclic codes
- 1.1, Definition:
injective, surjective, bijective
- 1.2, Definition:
permutation
- 1.3, Theorem:
The number of permutations of n letters.
- 1.3, Definition:
order of a permutation
- 2.1, Definition:
fixed points and support
- 2.2, Definition:
cycle, transposition
- 2.3, Proposition:
disjoint cycles form of a permutation
- 2.3, Definition:
cycle structure
- 2.4, Lemma:
conjugation
- 2.4, Proposition:
conjugacy is having the same cycle structure
- 2.5, Theorem:
each permutation is a product of transpositions
- 3.1, Theorem:
parity of the number of transpositions in a product expression
for a given permutation is constant
- 3.1, Definition:
sign of a permutation
- 3.2, Theorem:
sign is multiplicative
- 3.2, Corollary:
formula of the sign for a permutation in disjoint cycles form
- 3.3, Definition:
alternating group
- 3.3, Theorem:
the order of the alternating group
- 3.4, Theorem:
each even permutation is a product of 3-cycles
- 1.1, Definition:
unary operation, binary operation
- 1.2, Definition:
associativity
- 1.2, Theorem:
in compositions of an associative operation, bracketings not needed
- 1.3, Definition 1:
semi-group
- 1.3, Definition 2:
unit
- 1.4, Lemma:
a semi-group has at most one unit
- 1.4, Definition:
monoid
- 1.5, Definition:
commutativity
- 2.1, Definition:
direct product
- 2.2, Definition:
closed subset, submonoid
- 2.2, Theorem:
intersections of submonoids
- 2.3, Definition:
free monoid
- 2.4, Definition:
morphism, isomorphism
- 2.4, Theorem:
properties of isomorphisms
- 2.5, Theorem 1:
characterization of submonoid generated by a subset
- 2.5, Theorem 2:
characterization of cyclic monoids
- 3.1, Definition:
invertibility and inverse in a monoid
- 3.1, Theorem:
cancellation law
- 3.2, Theorem 1:
each element of a monoid has at most one inverse
- 3.2, Theorem 2:
properties of invertible elements in a monoid
- 3.3, Definition:
Euler indicator
- 3.3, Theorem:
recurrence relation for Euler indicator
- 3.3, Lemma:
towards a formula for the Euler indicator
- 4.1, Definition 1:
group
- 4.1, Definition 2:
product
- 4.1, Definition 3:
subgroup
- 4.2, Definition:
subgroup generated by a subset, cyclic group
- 4.3, Theorem:
intersections of subgroups are subgroups
- 4.3, Proposition:
characterization of a subgroup generated by a subset of a group
- 4.4, Proposition:
centralizer, normalizer, and center are subgroups
- 4.5, Definition:
morphism, isomorphism of groups
- 4.5, Proposition:
characterization of a morphism of groups
- 4.6, Theorem:
properties of morphisms
- 5.1, Theorem:
classification of cyclic groups
- 5.1, Definition:
order of a group element
- 5.2, Proposition:
properties of cyclic groups
- 5.2, Theorem:
characterization of finite cyclic groups
- 6.1, Lemma:
being equal up to right multiplication by a given subgroup is an equivalence relation
- 6.1, Definition:
cosets
- 6.2, Theorem:
Lagrange's theorem
- 6.2, Corollary 1:
the order of an element divides the order of the finite group to which it belongs
- 6.2, Corollary 2:
Fermat's little theorem
- 6.3, Theorem:
characterizations of normality of a subgroup
- 6.3, Proposition:
the kernel of a morphism is a normal subgroup
- 1.1, Definition 1:
ring
- 1.1, Definition 2:
subring
- 1.2, Theorem:
laws in rings involving 0 and 1
- 1.3, Theorem:
The invertible elements of a ring form a group
- 1.4, Definition:
morphism, kernel, image
- 1.4, Theorem:
properties of ring morphisms
- 2.1, Theorem 1:
Cartesian product of rings is a ring
- 2.1, Theorem 2:
the invertible elements in a Cartesian product form the product of
the invertible elements subgroup in each component
- 2.2, Theorem:
the intersection of subrings is a subring
- 2.2, Definition:
subring generated by a subset
- 2.3, Theorem:
a polynomial ring is a ring
- 3.1, Definition:
multiple, zero divisor, domain
- 3.1, Theorem:
properties concerning zero divisors
- 3.2, Theorem 1:
a polynomial ring is a domain if its coefficientring is a domain
- 3.2, Theorem 2:
cancellation law for domains
- 3.3, Definition:
field
- 3.3, Theorem:
every finite domain is a field
- 3.4, Definition:
subfield, subfield generated by a subset
- 3.4, Theorem:
characterization of certain subfields of C
- 3.5, Theorem:
the field of fractions of a domain is a field
- 4.1, Theorem:
the smallest subfield of a field is Q or Z/nZ
- 4.1, Definition:
characteristic of a field
- 4.2, Theorem:
A field containing a subfield is a vector space over that subfield
- 4.2, Corollary:
the order of a finite field is a prime power
- 4.3, Theorem:
the residue class of a polynomial ring with respect to an irreducible polynomial is a field
- 4.3, Lemma:
zeros and linear factors of a polynomial are related
- 4.4, Theorem:
properties of morphisms
- 4.4, Proposition:
properties of the morphism x -> xp in a field of
characteristic p.
- 4.5, Definition:
algebraic numbers
- 4.5, Theorem:
the algebraic numbers in C form a field
- 5.1, Definition:
ideal
- 5.1, Theorem:
characterization of an ideal generated by a finite subset
- 5.2, Theorem:
intersections of ideals are ideals
- 5.2, Proposition:
criteria for an ideal to be the whole ring
- 5.3, Proposition:
the sum of two ideals is an ideal
- 5.3, Theorem:
the kernel of a morphism is an ideal
- 5.4, Definition:
primality and maximality of an ideal
- 5.4, Theorem:
a maximal ideal is prime
- 6.1, Proposition:
congruence modulo an ideal is an equivalence relation
- 6.1, Theorem:
the residue class ring is a ring
- 6.2, Theorem:
first isomorphism theorem
- 6.3, Theorem:
characterization of primality and maximality in terms of quotient ring
- 7.1, Theorem 1:
Fermat's little theorem
- 7.1, Theorem 2:
characterization of finite fields
- 7.2, Lemma:
divisibility of polynomials whose zeros are roots of 1.
- 7.2, Theorem:
the multiplicative group of a finite field is cyclic
- 7.3, Theorem:
classification of finite fields
- 7.4, Theorem:
construction of Hadamard matrices
- 1.1, Definition:
permutation group, permutation representation
- 1.2, Proposition:
left multiplication, right multiplication, and conjugation are permutation representations
- 1.2, Theorem:
descriptions of the kernels of the permutation representations
by multiplication and by conjugation
- 1.3, Proposition:
constructions of permutation representations
- 2.1, Definition:
orbits, transitivity
- 2.1, Algorithm:
orbit construction
- 2.2, Definition:
stabilize, fix, stabilizer
- 2.2, Theorem:
left multiplication on cosets is a transitive permutation representation
- 2.3, Theorem:
a transitive representation is equivalent to left multiplication on cosets of stabilizer
- 3.1, Definition:
basis
- 3.1, Theorem:
order in terms of stabilizer orders and orbit lengths
- 3.2, Definition:
Schreier tree
- 3.2, Algorithm:
construction of a Schreier tree
- 3.3, Theorem:
generators for the stabilizer subgroup
- 3.3, Algorithm:
determining the order of a permutation group
- 4.1, Definition:
automorphism
- 4.1, Theorem:
the automorphisms forms group
- 4.2, Proposition:
field isomorphisms coming from root correspondence
- 4.2, Theorem:
description of the automorphism group of a finite field
- 5.1, Proposition:
properties of a normal subgroup
- 5.1, Definition:
quotient group
- 5.2, Proposition:
there is a morphism from a group to a quotient group
- 5.2, Theorem:
first isomorphism theorem for quotient groups
- 6.1, Definition:
dihedral group, quaternion group
- 6.1, Lemma:
existence of elements of prime order
- 6.2, Theorem:
classification of groups of order less than 12.
1