Let a be in Z[i], and suppose that N(a) = p^2 for some prime p in Z for which $\displaystyle p \equiv 3 $ (mod 4). Show that a is irreducible in Z[i].
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Originally Posted by tttcomrader Let a be in Z[i], and suppose that N(a) = p^2 for some prime p in Z for which $\displaystyle 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 $\displaystyle p \equiv 3 $ (mod 4) then p cannot be a sum of two squares.
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