Table of Gapplets
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Chapter 1. Arithmetic
- Divisors and multiples
- Euclid's algorithm
- Linear equations with integers
- Theorem page 1: Solving a Diophantine homogeneous equation
- Algorithm page 2: Solving a Diophantine equation
- Prime numbers
- Definition page 1: Test primality
- Theorem page 1: Find a prime not dividing a set of integers
- Algorithm page 2: Eratosthenes' sieve
- Factorization
Chapter 2. Modular arithmetic
- Computing modulo m
- Definition page 1: Congruence modulo m
- Definition page 2: Sum and product modulo m
- Definition page 3: Test invertibility modulo m
- Theorem page 3: Invertible elements modulo m
- Linear congruences
- Theorem page 1: Solve ax = b mod m
- Algorithm page 2: Solve ax = b mod m, describing all solutions explicitly
- The a-ary system
- Definition page 1: a-ary presentation of n
- Theorem page 1: From a-ary to b-ary presentation of a number
- The RSA-cryptosystem
- Theorem page 1: The powers of an integer modulo a prime
- Definition page 2: Find two primes and two keys for RSA encryption
- Definition page 3: RSA encryption
- Cool button: RSA encryption
- Radar detection
Chapter 3. Polynomials
- The polynomial ring
- Definition page 1: The terms, coefficients, and monomials of a polynomial
- Definition page 2: Sum and product in Z/nZ[X]
- Polynomial Division
- Definition page 1: The degree and leading coefficient of a polynomial
- Definition page 2: Divisibility of one polynomial by another
- Theorem page 2: Division for polynomials
- Definition page 3: Gcd and lcm for polynomials
- Algorithm 1 page 4: Gcd for polynomials
- Algorithm 2 page 4: Extended Euclidean algorithm for polynomials
- Polynomial functions
- Factorization
- Definition page 1: Test irreducibility of a polynomial
- Lemma page 1: The product of all distinct linear factors in Z/pZ[X]
- Theorem page 3: The set of irreducible factors of a polynomial
- Shift registers
- Definition page 1: The output of an integer shift register
- Theorem page 2: The output of an integer shift register as coefficients of a polynomial
- Fact page 3: The output of an integer shift register as coefficients of a polynomial
- Theorem page 3: The output of an integer shift register as coefficients of a polynomial
Chapter 4. Modular arithmetic for polynomials
- Congruence modulo a polynomial
- Definition page 1: Congruence of two polynomials modulo a third
- Theorem page 3: Remainder of division of one polynomial by another
- The residue class ring
- Definition page 1: The modular product, sum, and inverse
- Theorem page 4: The matrix of left multiplication by a polynomial in Q[X]/(d)
- Theorem page 5: The matrix of the projection map of polynomials onto Q[X]/(d)
- Two special cases
- Inverses and fields
- Theorem page 2: Invertible elements in Z/pZ[X]/(d)
- Corollary page 2: Find the inverse of an element of Z/pZ[X]/(d)
- Finite fields
- Theorem page 1: Find the order of an element in Z/pZ[X]/(d)
- Fact page 1: Find an irreducible polynomial of given degree in Z/pZ[X]
- Fact 1 page 2: Find a primitive element in Z/pZ[X]/(d)
- Cool button: The addition table of a field whose nonzero elements are given as powers of a primitive element
- Error correcting codes
- Definition page 1: Some simple encodings of a message
- Definition 1 page 2: The dimension of the row space of a given matrix over Z/pZ
- Definition 2 page 2: The dimension and minimal distance of the row space of a given matrix over Z/pZ
- Definition page 3: Constructing binary codes with factors of Xn - 1
- Theorem page 4: Encoding and decoding using cyclic codes
Chapter 5. Permutations
- The symmetric group
- Definition page 1: Injectivity test of a simple map
- Definition page 2: Product of permutations given in list form
- Theorem page 3: Order of the symmetric group, and some elements
- Definition page 3: The order of a permutation
- Cycles
- Definition page 1: Fixed points and support of a permutation
- Definition page 2: From disjoint cycles form to list form
- Proposition page 3: From list form to disjoint cycles form
- Definition page 3: The cycle structure of a permutation
- Lemma page 4: Conjugation
- Proposition page 4: The order of Sn, some elements
- Theorem page 5: Write a permutation as a product of 2-cycles
- The alternernating group
Chapter 6. Monoids and groups
- Semigroups
- Definition page 2: Test associativity of a binary operation Z×Z -> Z given by a polynomial in Z[x,y]
- Definition page 3: Composition in the semigroup Mn
- Definition page 5: Given polynomial in Z[x,y], check if it defines a commutative binary operation on Z
- Monoids
- Definition page 3: The number of balanced bracketings of given length
- Definition page 4: Given a map on {1, ...,n} in list form, determine its image on pairs
- Theorem 1 page 5: Given a sequence of maps in Mn, determine the submonoid they generate
- Theorem 2 page 5: Given a map in Mn, determine the cyclic submonoid it generates
- Invertibility in monoids
- Theorem 2 page 2: Given a sequence of maps in Mn, determine the invertible elements of the submonoid they generate
- Definition page 1: Given a map in Mn, decide if it is invertible.
- Definition page 3: The Euler indicator
- Groups
- Definition page 1: Given a sequence of maps in Mn, determine whether they generate a group
- Definition page 2: Given a sequence of permutations Sn, determine the order of the subgroup they generate
- Theorem 2 page 3: The intersection of two subgroups of Sn
- Theorem 1 page 4: The centralizer, normalizer and center of a subgroup of S5
- Cyclic groups
- Cosets
Chapter 7. Rings and fields
- The structure ring
- Definition page 1: Test distributivity
- Theorem page 3: The Euler indicator
- Definition page 4: Given a square matrix, determine its minimum polynomial and inverse
- Cool button: The multiplication table of (Z/mZ)*
- Constructions with rings
- Theorem 2 page 1: Determine the structure of the multiplicative group (Z/mZ)*
- Theorem page 3: Arithmetic in Z[x,y]
- Domains and fields
- Definition page 1: Given a matrix, check if it is a zero divisor and if it is invertible
- Definition page 4: Subfield of finite fields
- Theorem page 5: Arithmetic in Q(X), the field of fractions of Q[X]
- Fields
- Theorem page 2: The matrix of multiplication by f in Q[X]/(d)
- Lemma page 3: Linear factors of a polynomial in Q[X]
- Definition page 5: Determine the minimum polynomials of some special algebraic numbers
- Theorem page 5: Determine the minimum polynomial of the sum and product of two algebraic numbers
- Cool button: Determine the minimum polynomials of some special algebraic numbers
- Ideals
- Residue class rings
- Finite fields
Chapter 8. Permutation groups
- Permutation representations
- Definition page 1: The permutation on nonzero vectors determined by a 2×2 matrix over Z/pZ
- Proposition page 2: Conjugation by a permutation on elements of given cycle structure viewed as a permutation
- Theorem page 2: The center of a group of permutations described in terms of the generators of the group
- Orbits
- Definition page 1: Orbits of a permutation group
- Algorithm page 1: Orbit of a point in a group, the algorithm step by step
- Definition page 2: The stabilizer subgroup
- Theorem page 2: The correspondence between an orbit and the left cosets of a point stabilizer
- Theorem page 3: From an orbit to a collection of cosets
- Order
- Definition page 1: Basis for a permutation group
- Theorem page 1: Basis for a group and corresponding order computation
- Definition page 2: Schreier tree for a permutation group
- Algorithm page 2: Schreier words for a permutation group
- Algorithm page 3: Stabilizer of point and its order
- Theorem page 3: Stabilizer of a point and Schreier words
- Automorphisms
- Definition page 1: Permutation corresponding to the map x -> xp on (Z/pZ)*
- Theorem page 1: The automorphism group of a small graph
- Theorem page 2: An automorphism of a finite field written as a permutation
- Quotient groups
- Small groups