Chapter 6 definitions

Question 1

coprime

Answer 1

Two elements of R are coprime if their only common factors are units.

Question 2

ordₚ(a)

Answer 2

Let a∈R{0} and let p∈R be irreducible. Define, ordₚ(a) to be the number of associates of p that occur in a factorisation of a into irreducibles.

Question 3

Method of descent

Answer 3

To show f has no non-trivial integer solutions we need
1. a formula for measuring the size of a solution
2. a machine which when given a solution produces another smaller solution
The machine produces infinitely many solutions which is a contradiction since only a finite number of non-negative integers less than a.

Question 4

Mod in any ring

Answer 4

Let R be a ring and n̸=0, α,β∈R define α≡βmodn to mean n|(α-β) in R.

Question 5

CSR

Answer 5

Let R be a ring and 0 ̸= n ∈ R. A complete set of residues mod n in R is a set S of elements of R such that each member of R is congruent modulo n to exactly one member of S