Consider the integral domain .

Show that and have no common factors in except for and .

I can show that both and are irreducible. However, is there another way to prove that and are the only common factors?

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- Jun 8th 2012, 07:14 AMmath2011Common factors in integral domain Z[sqrt(-13)]
Consider the integral domain .

Show that and have no common factors in except for and .

I can show that both and are irreducible. However, is there another way to prove that and are the only common factors? - Jun 8th 2012, 01:18 PMwsldamRe: Common factors in integral domain Z[sqrt(-13)]
Showing irreducibility is not enough, you must also prove that the only units in your domain are 1 and -1.

- Jun 8th 2012, 06:01 PMmath2011Re: Common factors in integral domain Z[sqrt(-13)]
But should showing irreducibility be the first step though? Or is there a different solution that does not involve proving irreducibility?

- Jun 8th 2012, 07:45 PMmath2011Re: Common factors in integral domain Z[sqrt(-13)]
Suppose is a common factor.

Then for some and . It can be deduced that or .

We also have for some and . It can be seen that is the only possible value.

Hence is the only common factor.