Every prime congruent to 1 mod 4 is a sum of squares

Prove that every prime integer p congruent to 1 mod 4 is a sum of two squares. Show that every product of two such primes is also a sum of two squares.


Let p be such a prime. We know that p is not irreducible in \mathbb{Z}[i]; thus there exist nonunits \alpha,\beta such that p = \alpha\beta. Note that neither of \alpha and \beta can have norm 1. Since p^2 = N(p) = N(\alpha)N(\beta), letting \alpha = a+bi, we see that a^2 + b^2 = p as desired.

Now suppose p,q are integer primes congruent to 1 mod 4. As above, we have p = \alpha_1\beta_1 and q = \alpha_2\beta_2, where \alpha_i and \beta_i have norm p and q as needed. Now N(\alpha_1\beta_2) = pq. In particular, if \alpha_1\beta_2 = a+bi, then a^2+b^2 = pq.

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