Let be an integral domain. A Euclidean valuation is a function from the nonzero elements of to the nonnegative integers such that the following hold:

• For any with , there exist such that with or .
• For any , we have .

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 is . That is, for any .
• is a unit if and only if .
• For any and any unit of , we have .

These facts can be proven as follows:

• If , then
  
• If is a unit, then let be its inverse. Thus,
  
  Conversely, if , then there exist with or such that
  
  Since is impossible, we must have . Hence, is the inverse of .
• Let be the inverse of . Then
  

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 is a Euclidean domain under the Euclidean valuation for all .
• is a Euclidean domain with absolute value acting as its Euclidean valuation.
• If is a field, then , the ring of polynomials over , is a Euclidean domain with degree acting as its Euclidean valuation: If is a nonnegative integer and with , then
  

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.