Let be the Euclidean norm given by .
- Show that for all .
- Show that if and only if is a unit.
- Show that if and only if .
- Show that if is prime as a natural number then is irreducible in .
- Show that and are prime.
Let and , and note the following.
Suppose is a unit; then there exists such that . Considering norms, we have (using the previous result). So . Conversely, suppose ; then , and we have . Thus is a unit.
Certainly . Now suppose ; then .
If is prime and , then is prime. Without loss of generality, , so that is a unit. Thus is irreducible.
Note that is prime; thus is irreducible.
Note that has no solutions in the integers since 3 is not a perfect square. In particular, no element of has norm 3. Now . If , then . since no element of this ring has norm 3, without loss of generality we have so that is a unit. Thus is irreducible.