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Problem statement:

Find a generator of the ideal $(85, 1+13i)$ in $\mathbb{Z}[i]$, i.e., a GCD for $85$ and $1 + 13i$ by the Euclidean Algorithm. Do the same for the ideal $(47-13i, 53+56i).$

Can you please outline the steps, then I can practice with others.


Source: Abstract Algebra by Dummit & Foote, $\S$8.1 #7

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    Euclid algorithm http://en.wikipedia.org/wiki/Euclidean_algorithm - your remainders should have the minimal absolute value2011-04-10
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    Since $85 = 5\times 17 = (1+2i)(1-2i)(1+4i)(1-4i)$ and $(1+13i)(1-13i) = 170 = 2\times 5\times 17 = (1+i)(1-i)(1+2i)(1-2i)(1+4i)(1-4i)$, it is now easy to check that $1+13i = -i(1+i)(1+2i)(1+4i)$, so that $\gcd(85,1+13i)=(1+2i)(1+4i) = -7+6i$.2011-04-10
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    I resurrected this question (by editing it), since I came across the problem in an algebra book.2011-11-11
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    It's also worth pointing out (although not worth posting as an official answer) that since $\mathbb{Z}[i]$ is a UFD, you can compute the $gcd$ of any two nonzero nonunit elements by looking at the prime factorizations in $\mathbb{Z}[i]$ (by, eg, using the norm). Of course, this is computationally much more difficult than simply applying the Euclidean algorithm.2011-11-11

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