First define the conjugate of to be . Also define the norm of to be .

An algebraic integer in the field is of the form , where . If a number is an algebraic integer, its norm is an ordinary integer.

Write as , with , , and integers.

So .

Suppose that and are algebraic integers in . In the field , the quotient can be written as , where and are rational. Now we want to get the quotient in the ring of algebraic integers. The procedure is this: Let , where and are ordinary integers such that and . Then . The reminder is . So we write , where . The button "division in integers" shows the pair , the quotient and reminder.