# Math Help - Mod Irreducible

1. ## Mod Irreducible

Let a be in Z[i], and suppose that N(a) = p^2 for some prime p in Z for which $p \equiv 3$ (mod 4). Show that a is irreducible in Z[i].

Let a be in Z[i], and suppose that N(a) = p^2 for some prime p in Z for which $p \equiv 3$ (mod 4). Show that a is irreducible in Z[i].
If a = bc and neither b nor c is a unit, then N(b) = p. But if $p \equiv 3$ (mod 4) then p cannot be a sum of two squares.