Let N be a multiplicative norm such that (verify this is an indeed multiplicative norm).

We have for all in , and if is a unit.

Now consider 2 and suppose 2= .

Then . If 2 is not irreducible, should be 2, but there is no integer satisfyng . Contradiction! Thus 2 is irreducible in .

Now consider and suppose .

Then . If is not irreducible, then should be either 2 or 3. But there is no integer satisfyng or 3. Contradiction! Thus is irreducible in .

I leave it to you to show the remainder elements are irreducible in .