Chapter 1 definitions

Question 1

Gaussian Integer

Answer 1

A Gaussian integer is a complex number whose real and imaginary parts are both integers.

Question 2

Ring

Answer 2

A set R of complex numbers is a ring if

(i) x+y, x-y, xy ∈R for all x,y∈R
(ii) 0,1∈R

Question 3

Factor

Answer 3

Let a,b be elements of a ring R. We say that a is a factor of b in R, a|b, if there exists z∈R s.t. b=az

Question 4

Unit

Answer 4

An element u of a ring R is a unit in R if u|1 in R. That is if 1/u ∈ R.

Question 5

Associates

Answer 5

Let x be an element of a ring R. The associates of x are the elements ux where u is a unit of R.

Question 6

Irreducible

Answer 6

A non-zero non-unit element x of a ring R is irreducible in R if the only factors of x in R are the associates of x and the units of R

Question 7

Euclidean Ring

Answer 7

A ring R of complex numbers is a Euclidean Ring (ER) if

(i) |a|²∈ℤ for all a∈R
(ii) given any fraction of the form z=a/d with a,d∈R, d not 0. Then there exists a q∈R s.t. |z-q|²<1
i. e. z has distance less than 1 some some member of R

Question 8

Highest Common Factor

Answer 8

Let R be a ring and a,b∈R. A highest common factor is an element h∈R with the properties
(i) h|a, h|b
(ii) for each c∈R if c|a and c|b then c|h (if c is a common factor then then c must divide h)
HCF(a,b) denotes the set of all highest common factors

Question 9

prime

Answer 9

Let R be a ring. A non-zero non-unit element p∈R is prime if for all a,b∈R we have that p|ab => p|a or p|b

Question 10

Unique Factorisation Domain (UFD)

Answer 10

A UFD is a ring R with the property that for each a∈R with a a non-zero non-unit

(i) There is exists irreducibles p₁, … , pₙ s.t. a=p₁…pₙ
(ii) If q₁,…,qₘ are also irreducibles with a = q₁…qₘ then m=n and there is a 1-1 correspondence between the ps and the qs s.t. corresponding numbers are associate.