Talk:Gaussian integers

There is a proof that the Gaussian integers form a unique factorization domain in the Fine & Rosenberger book but that proof requires an understanding of Euclidean norms and domains. I wonder if there is an effective proof using less advanced concepts? Alonso del Arte 00:39, 23 May 2012 (UTC)

From LeVeque, p. 102, section 6-4, Theorem 6-5. Every integer alpha with Norm alpha > 1 can be represented as a finite product of primes. Theorem 6-6. If alpha divides beta gamma and gcd(alpha, beta) = 1, then alpha divides gamma. Theorem 6-7. If pi, pi_1, ... pi_n are GAussian primes, and pi divides pi_1 * ... * pi_n, then at least for one m, pi is an associate of pi_m. Theorem 6-8 (Unique Factorization Theorem for Z[i]). The representation of each Gaussian integer alpha with Norm alpha > 1 as a product of primes is unique except for the order of factors and the presence of units.

Proof for 6-8: Suppose there is an alpha with two factorizations. By 6-7, pi_1 divides pi'_1 * ... * pi's. Cancelling pi_1 we obtain alpha/pi_1 = ah, ran out of time. Alonso del Arte 23:07, 15 May 2014 (UTC)