I'm trying to factorise the Gaussian integer $z =11 - 3i$ into Gaussian primes. Taking the Euclidean norm on $z$ is $\nu (z) = 130$ which factorises into $2 \times 5 \times 13$ and so I'm assuming I want to find Gaussian primes $a, b, c$ such that $\nu(a) = 2,\ \nu(b) =5,\ \nu(c) = 13$ ? But I can't see how to go about doing this? Am I on the right track or is there a set method to factorise into gaussian primes?
Factorising into Gaussian primes
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abstract-algebra
prime-numbers
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0which numbers? $2, 5, 13$? – 2012-06-03
1 Answers
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Hint: Try $(1+i)(1-i)$, $(2+i)(2-i)$, and $(3+2i)(3-2i)$.
Useful Fact: a number is a sum of two squares if and only if each prime factor of that number that is equal to $3\pmod{4}$ occurs with even exponent.
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0Ah! I understand how they're linked now, thank you – 2012-06-03