Chapter 3 definitions

Question 1

a is congruent to b modulo n

Answer 1

Let a,b∈ℤ and n∈ℕ. We write a≡bmodn and say a is congruent to b modulo n if n|(a-b)

Question 2

Complete Set of Residues

Answer 2

Let n∈ℕ. The a complete set of residues (CSR) modulo n is a subset S⊆ℤ with the property that every Integer is congruent modulo n to exactly one element of S.

Question 3

polynomial modulo n

Answer 3

let f(X) g(X) be polynomials with integer coefficients and let n∈ℕ. We say that f(X) is congruent to g(X) if for each i≥0 the coefficients of xᶦ if f(X) are congruent modulo n to the coefficiant of xᶦ in g(X).

Question 4

degree of f(X) modulo n

Answer 4

Suppose that n∈ℕ and that f(X) is a polynomial with integer coefficients. The degree modulo n of f(X) is the largest d such that the coefficient of Xᵈ is not congruent to 0 modulo n. If all the coefficients are congruent to 0 modulo n then we define the degree to be -∞

Question 5

order of a modulo p

Answer 5

Let p be prime and a∈ℤ with a≢0modp then the order of a mod p is the smallest natural number d∈ℕ s.t. aᵈ≡1modp

Question 6

primitive rot modulo p

Answer 6

suppose that p is prime. A primitive root modulo p is an integer ω with order p-1 modulo p.