## Euclidean Domains

Definition 1   An integral domain endowed with a function is an Euclidean domain if the following two conditions hold
• for all with and we have ,
• for all with there exist such that

 (1)

The elements and are called the quotient and the remainder of w.r.t. (although and may not be unique). The function is called the Euclidean size.

Example 1   Here are some classical examples.
• with for . Here the quotient and the remainder of w.r.t. (with ) can be made unique by requiring (hence we have ).
• where is a field with the degree of for and . Uniqueness of the quotient and the remainder is easy to show in that case. Indeed

 (2)

implies

 (3)

Hence we must have and thus .
• is a field with for and . In this case the quotient and the remainder of w.r.t. are and 0 respectively.
• Let be the ring of the complex numbers whose real and imaginary parts are integer numbers. Hence

 (4)

Consider as a map from to the norm of an element. Hence with . It is easy to check that for every with we have . Indeed for we have

 (5)

Moreover for every we have . Therefore we have proved that holds for every with . Now given with we are looking for a quotient and a remainder of w.r.t. . Hence we are looking for such that . Such a can be constructed as follows. Let be such that that is where is the conjugate of . Hence writes with . Let be such that and . Then

 (6)

It turns out that several can be chosen. For instance with and we have with and with . In both cases . Finally this shows that a quotient and a remainder of w.r.t. may not be uniquely defined in .

Marc Moreno Maza
2008-01-07