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I am trying to solve this problem:

Find the generator of the ideal $(47 - 13i, 53 + 56i).$ I know that I should use Euclidean Algorithm but I am wondering if it matters if I divided $a = 47 - 13i$ by $b = 53 + 56i$ or $b = 53 + 56i$ by $a = 47 - 13i$? and why?

Thanks for your help!

Peter Phipps
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Brain
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    You can divide in either order because $\gcd(a,b) = \gcd(b,a)$. – paw88789 Nov 04 '21 at 22:47
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    You won't get a reduction to a "smaller" gcd problem if you don't choose the smallest argument as the divisor - just like for integers. But that will be corrected in the next step when you swap the arguments. The same is true in any (algorithmic) Euclidean domain, e.g. for polynomials over a field- where the size measure is "degree". – Bill Dubuque Nov 04 '21 at 22:52
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    @Brain I agree with Ethan's answer. – paw88789 Nov 04 '21 at 22:58

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When you use the Euclidean algorithm for ordinary integers you divide by the smaller (in absolute value) number in order to get a small remainder for the next step.

The algorithm would still work if you did it the other way since the first step would have a quotient of $0$ and a remainder the smaller number.

In the Gaussian integers you measure size using the norm.

Take it from there ...

Ethan Bolker
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  • Oh, thank you that is great! – Brain Nov 04 '21 at 22:51
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    The comment on the questoin is correct. I edited my answer to address it. @BillDubuque said the same thing in his comment there. – Ethan Bolker Nov 04 '21 at 22:55
  • Sorry for bothering you, but is it the greatest common divisor or just a greatest common divisor that we will find i.e., is gcd unique or not? – Brain Nov 04 '21 at 23:19
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    It's a gcd. In the integers $-3$ works just as well as $3$ for $\gcd(6,9)$. In the Gaussian integers you multiply any gcd you find by $\pm 1$ and $\pm i$ to find them all. The idea you will learn is "unique up to multiplication by a unit". – Ethan Bolker Nov 04 '21 at 23:23
  • Perfect! thank you :)! – Brain Nov 04 '21 at 23:28
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    @Brain Generally gcds are unique only up to associates (-= unit multiples in domain), see here. The quotient and remainder in the Euclidean algorithm are unique only in a polynomial ring over a field (or a field), see here – Bill Dubuque Nov 04 '21 at 23:29
  • @BillDubuque thank you very much for the very detailed reply :)! – Brain Nov 04 '21 at 23:30