In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity).
In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra.
[2] Some authors also require the domain of the Euclidean function to be the entire ring R;[2] however, this does not essentially affect the definition, since (EF1) does not involve the value of f (0).
Note that, for a Euclidean function that is so established, there need not exist an effective method to determine q and r in (EF1).
[12] In particular this applies to the case of totally real quadratic number fields with trivial class group.
In addition (and without assuming ERH), if the field K is a Galois extension of Q, has trivial class group and unit rank strictly greater than three, then the ring of integers is Euclidean.
[13] An immediate corollary of this is that if the number field is Galois over Q, its class group is trivial and the extension has degree greater than 8 then the ring of integers is necessarily Euclidean.
Algebraic number fields K come with a canonical norm function on them: the absolute value of the field norm N that takes an algebraic element α to the product of all the conjugates of α.