Let $D$ be an integral domain. A Euclidean valuation is a function from the nonzero elements of $D$ to the nonnegative integers $\nu \colon D \setminus \{0_D\} \to \{ x \in \mathbb{Z} : x \ge 0 \}$ such that the following hold:

• For any $a,b\in D$ with $b\neq 0_D$ , there exist $q,r\in D$ such that $a=bq+r$ with $\nu(r)<\nu(b)$ or $r=0_D$ .
• For any $a,b\in D \setminus \{0_D\}$ , we have $\nu(a)\leq\nu(ab)$ .

Euclidean valuations are important because they let us define greatest common divisors and use Euclid's algorithm. Some facts about Euclidean valuations include:

• The minimal value of $\nu$ is $\nu(1_D)$ . That is, $\nu(1_D)\leq\nu(a)$ for any $a\in D \setminus \{0_D\}$ .
• $u\in D$ is a unit if and only if $\nu(u)=\nu(1_D)$ .
• For any $a\in D \setminus \{0_D\}$ and any unit $u$ of $D$ , we have $\nu(a)=\nu(au)$ .

These facts can be proven as follows:

• If $a\in D \setminus \{0_D\}$ , then$$ \nu(1_D)\leq\nu(1_D\cdot a)=\nu(a).$$
• If $u\in D$ is a unit, then let $v\in D$ be its inverse. Thus,$$ \nu(1_D)\leq\nu(u)\leq\nu(uv)=\nu(1_D).$$ Conversely, if $\nu(u)=\nu(1_D)$ , then there exist $q,r\in D$ with $\nu(r)<\nu(u)=\nu(1_D)$ or $r=0_D$ such that$$ 1_D=qu+r.$$ Since $\nu(r)<\nu(1_D)$ is impossible, we must have $r=0_D$ . Hence, $q$ is the inverse of $u$ .
• Let $v\in D$ be the inverse of $u$ . Then$$ \nu(a)\leq\nu(au)\leq\nu(auv)=\nu(a).$$

Note that an integral domain is a Euclidean domain if and only if it has a Euclidean valuation.

Below are some examples of Euclidean domains and their Euclidean valuations:

• Any field $F$ is a Euclidean domain under the Euclidean valuation $\nu(a)=0$ for all $a\in F \setminus \{0_F\}$ .
• $\mathbb{Z}$ is a Euclidean domain with absolute value acting as its Euclidean valuation.
• If $F$ is a field, then $F[x]$ , the ring of polynomials over $F$ , is a Euclidean domain with degree acting as its Euclidean valuation: If $n$ is a nonnegative integer and $a_0,\dots,a_n\in F$ with $a_n\neq 0_F$ , then$$ \nu\left(\sum_{j=0}^n a_jx^j\right)=n.$$

Due to the fact that the ring of polynomials over any field is always a Euclidean domain with degree acting as its Euclidean valuation, some refer to a Euclidean valuation as a degree function. This is done, for example, in Joseph J. Rotman's A First Course in Abstract Algebra.