nLab
Euclidean domain

Contents

Idea

A Euclidean domain is an integral domain which admits a form of the Euclidean quotient-and-remainder algorithm familiar from school mathematics.

Definition

A Euclidean domain is an integral domain AA for which there exists a function d:A{0}d: A \setminus \{0\} \to \mathbb{N} to the natural numbers, often called a degree function, such that given f,gAf, g \in A with g0g \neq 0, there exists unique q,rAq, r \in A such that f=qg+rf = q \cdot g + r and either r=0r = 0 or d(r)<d(g)d(r) \lt d(g). (One may harmlessly stipulate that d(0)=0d(0) = 0; what to do with the zero element varies from author to author.)

Some authors also add the requirement that d(a)d(ab)d(a) \leq d(a b) for all nonzero a,ba, b. There is no loss of generality in assuming it; every Euclidean domain admits such a degree function dd', defining d(a)=min{d(ab):bA,b0}d'(a) = \min \{d(a b): b \in A, b \neq 0\}. We’ll use it freely below, if and when we need to.

Examples

  • The (rational) integers \mathbb{Z}.

  • The ring of polynomials k[x]k[x] over a field kk, using the ordinary polynomial degree.

  • Any field (trivially).

  • Any discrete valuation ring: letting π\pi be a generator of the maximal ideal, put d(x)=nd(x) = n where x=uπ nx = u \pi^n, with uu a unit.

  • The Gaussian integers [i]\mathbb{Z}[i], with d(a+bi)=a 2+b 2d(a + b i) = a^2 + b^2 (the norm).

  • The Eisenstein integers? [ω]\mathbb{Z}[\omega], where ω\omega is a primitive cube root of unity, with d(a+bω)=a 2ab+b 2d(a + b \omega) = a^2 - a b + b^2 (the norm).

Properties

Proposition

A Euclidean domain is a principal ideal domain.

Proof

Let IAI \subseteq A be a nonzero ideal, and suppose d(g)d(g) is the minimum degree taken over all nonzero gIg \in I. For such a gg and any fIf \in I, we may write f=qg+rf = q g + r where either r=0r = 0 or d(r)<d(g)d(r) \lt d(g) (which is impossible since rIr \in I and d(g)d(g) is minimal). So r=0r = 0 it is, and thus I=(g)I = (g).

Last revised on July 4, 2015 at 16:53:38. See the history of this page for a list of all contributions to it.