Euclidean Domain - Ring Theory, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

A Euclidean domain is an integral domain R which can be equipped with a function

d : R\{0} → N

such that for all a ∈ R and b ≠ 0, b ∈ R we can write

a = qb + r

for some q, r ∈ R with r = 0 or d(r) < d(b).

For example. Z with d(n) = |n| is a Euclidean Domain; also, for any field k, k[X ] with d(f ) = deg(f ) is a Euclidean Domain. (WARNING: In the second example above, it is essential that k be a field.)

We shall prove that every Euclidean Domain is a Principal Ideal Domain (and so also a Unique Factorization Domain). This shows that for any field k, k[X ] has unique factorization into irreducibles. As a further example, we prove that  is a Euclidean Domain.

Proposition 1. In a Euclidean domain, every ideal is principal. 

Proof. Suppose R is a Euclidean domain and  . Then EITHER I = {0} = (0) OR we can take a ≠ 0 in I with d(a) least; then for any b ∈ I , we can write b = qa + r with r = 0 or d(r) < d(a); but r = q − ba ∈ I and so by minimality of d(a), r = 0; thus a|b and I = (a).

Corollary 2. If k is a field then every ideal in k[X ] is principal.

Corollary 3. Let k be a field. Then every polynomial in k[X ] can be factorized into primes=irreducibles, and the factorization is essential ly unique.

Corollary 4. Every element of the ring  can be factorized into primes= irreducibles, and the factorization is essential ly unique.

Proof. By Theorem 1, it is enough to show that  is a Euclidean Domain.
To this end, define N :  → N by

Note that we can extend N to a function N : Q  → Q defined similarly by

Note also that given any  we have

Now, suppose we are given 

Then

Pick