Abbreviation: EucDom

A \emph{Euclidean domain} is an integral domains $\langle D,+,-,0,\cdot,1\rangle$ together with a function $d:D\setminus\{0\} \to\mathbf{N}$ such that

$\forall a,b\ (a\ne 0$, $b\neq 0 \Longrightarrow d(a)\le d(ab))$

$\forall a,b \exists q,r\ (a=b\cdot q+r$, $(r=0 \mbox{or} d(r)<d(b)))$

Example 1: $\langle\mathbb{Z},+,-,0,\cdot,1,d\rangle$, the ring of integers with addition, subtraction, zero, and multiplication is a Euclidean domain with $d(a)=|a|$.

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= &1 f(5)= &1 f(6)= &0 \end{array}$

Fields

Principal Ideal Domains