You have already seen a few domain extensions of integers of the form , where is a square-free integer (i.e., ). The case of our interest in this post are Gaussian integers , that are obtained by setting .
We will now study prime elements of and we will see whether it has some interesting consequences. The following statement is very useful in this situation:
Let be a prime number. If is NOT of the form for some , then is a prime element in . On the other hand, if , then and are prime elements. Moreover, up to multiplication by units, all prime elements of are of one of these types.
A very important consequence of this lemma is the following theorem:
A prime number can be written as or mod . All prime elements in (up to multiplication by units) are
- prime numbers, such that mod ;
- , where mod .
- is a Euclidian domain for , but it is not Euclidian for .
- There is an operation of conjugation in that takes an element and sends it to the element .
- We have a Euclidian norm on : if , then
- It is a general fact that is invertible. Therefore, units (invertible elements) of Gaussian integers are and .
Decide whether or are prime elements in .
We will use Lemma from above:
- , hence, is not a prime element , but and are prime.
- , since mod . Therefore, is a prime element.
- , so is not a prime element, but and are.
Show that and are the only invertible elements in .
We use the general algebraic fact mentioned above: is invertible . So, we are trying to solve an equation , where Since the square of any integer is non-negative, the only possible solutions are & , or & , and we are done.
Prove that and are associated in .
Compute the norm of the following elements: , and .