Thread: Show that a prime is not irreducible in Z[i]

Let p be prime with , show that p is not irreducible in Z[i].

Proof. Now from the previous problem, I know that p \ n^2 + 1 = (n+i)(n-i).

If p is irreducible, then p is a prime and it divides either (n+i) or (n-i). I'm trying to show that is impossible, any hint?

Thank you.

Originally Posted by tttcomrader

Let p be prime with , show that p is not irreducible in Z[i].

Proof. Now from the previous problem, I know that p \ n^2 + 1 = (n+i)(n-i).

If p is irreducible, then p is a prime and it divides either (n+i) or (n-i). I'm trying to show that is impossible, any hint?

Thank you.

By your other problem if then for some . Thus, , assume that is irreducible then or . If then by definition for some . Thus, which is an impossibility. Similarly, cannot divide . Thus, the assumption that is irreducible is false.