# Prime and Irreducible in Z[i]

• January 17th 2008, 07:55 PM
Prime and Irreducible in Z[i]
Let a be in Z[i], show that if N(a) is prime in Z, then a is irreducible in Z[i].
• January 17th 2008, 08:09 PM
ThePerfectHacker
Assume that $a = bc$ then $N(a) = N(b)N(c)$ if $N(a)$ is prime then one of $N(b)$ or $N(c)$ is $1$ since the norm is $1$ it means the element is a unit. And so $b$ or $c$ is a unit. Thus $a$ is irreducible.