Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 2 / 2E / i / Solution

A Euclidean domain is a commutative integral domain $R$ with a function $\delta :R\setminus \{0\}\to {\mathbb{Z}}_{\ge 0}$ such that, for $a,b\in R$ with $b\ne 0$, there are $q,r\in R$ satisfying $a=qb+r$, where either $r=0$ or $\delta (r)<\delta (b)$. One may also impose the usual monotonicity $\delta (a)\le \delta (ab)$ for nonzero $a,b$; the example below has it.

The integers form a Euclidean domain that is not a field. Use $\delta (a)=|a|$. Integer division by $|b|$ supplies a remainder with $0\le r<|b|$, and changing the sign of the quotient handles negative $b$. The integers are an integral domain, while $2$ has no multiplicative inverse in them. This verifies both the Euclidean property and the failure to be a field.