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I'm trying to find the GCD of $4+22i$ and $17+i$ in $Z[i]$. I started by using Euclid's and did the following steps:

$4 + 22i = (17 + i) \cdot i + (5 + 5i)$

$17 + i = (5 + 5i) \cdot (1 - i) + (7 + i)$

Now when I try to divide $5 + 5i$ by $7 + i$ I get

$\frac{5 + 5i}{7 + i} = \frac{5 + 5i}{7 + i} \cdot \frac{7 - i}{7 - i} = \frac{40}{50} + \frac{30i}{50}$

I'm not sure where to go from here (or if I've made a mistake along the way) because this would seem to give a quotient of $0$ which will cause the algorithm to loop.

UBears
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    Check the second division. At first glance the remainder seems to have the same squared length ($50$) as the quotient. It should be smaller. Did you find the nearest lattice point? – Ethan Bolker Mar 10 '17 at 01:14
  • So instead of choosing $1-i$ for the quotient for the second equation, would I want to choose $2 - 2i$? – UBears Mar 10 '17 at 01:27
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    Maybe/probably. I didn't try to solve the problem, I just noticed the sizes since $25+25 = 49 + 1$. Try it. If it works you can answer your own question. – Ethan Bolker Mar 10 '17 at 01:36
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    Try using the Gaussian version. – Nairit Mar 10 '17 at 01:52

2 Answers2

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Write $z=4+22i$ and $w=17+i$. Now, $$\frac{z}{w}=\frac{4+22i}{17+i}\cdot \frac{17-i}{17-i}=\frac{9}{29}+\frac{37}{29}i.$$ The closest integers to $\frac{9}{29}$ and $\frac{37}{29}$ are $0$ and $1$, respectively. Write $q_1=0+1i=i.$ Then $$r_1=(4+22i)-(17+i)(i)=5+5i.$$ Now, $$\frac{17+i}{r_1}=\frac{17+i}{5+5i}\cdot \frac{5-5i}{5-5i}=\frac{9}{5}+\frac{-8}{5}i.$$ The closest integers to $\frac{9}{5}$ and $\frac{-8}{5}$ are $2$ and $-2$, respectively. Write $q_2=2-2i$. Then $$r_2=(17+i)-r_1q_2=(17+i)-(5+5i)(2-2i)=-3+i.$$ Now, $$\frac{r_1}{r_2}=\frac{5+5i}{-3+i}\cdot\frac{-3-i}{-3-i}=-1-2i=q_3\in\Bbb{Z}[i].$$ We have shown that $$\begin{align} z&=wq_1+r_1\\ w&=r_1q_2+r_2\\ r_1&=r_2q_3+0. \end{align}$$ The last nonzero remainder is $r_2$ and hence (using the euclidean algorithm) the required $\gcd$ is $r_2=-3+i$.

Juniven Acapulco
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As Ethan Bolker pointed out in the comments, I chose an incorrect divisor and should have thought about the nearest lattice point. Here is the correct progression:

$4 + 22i = (17 + i) \cdot i + (5 + 5i)$

$17 + i = (5 + 5i) \cdot (2 - 2i) + (-3 + i)$

$5 + 5i = (-3 + i) \cdot (1 - 2i) + 0$

So GCD of $4+22i$ and $17+i$ is $-3 + i$.

UBears
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