The document Euclidean Domain Mathematics Notes | EduRev is a part of the Mathematics Course Algebra for IIT JAM Mathematics.

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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

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