Common factors in integral domain Z[sqrt(-13)]
Consider the integral domain ![\mathbb{Z}\[\sqrt{-13}\] = \{ x + y \sqrt{-13} | x,y \in \mathbb{Z} \}](http://latex.codecogs.com/png.latex?\mathbb{Z}\[\sqrt{-13}\] = \{ x + y \sqrt{-13} | x,y \in \mathbb{Z} \})
.
Show that 
and 
have no common factors in ![\mathbb{Z}\[\sqrt{-13}\]](http://latex.codecogs.com/png.latex?\mathbb{Z}\[\sqrt{-13}\])
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?
Re: 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.
Re: 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?
Re: Common factors in integral domain Z[sqrt(-13)]
Suppose 
is a common factor.
Then 
for some 
and  = 2^2 = N(\alpha)N(x) = (a^2 + 13 b^2)(c^2 + 13d^2))
. It can be deduced that 
or 
.
We also have 
for some 
and  = 22 = N(\alpha)N(y) = (a^2 + 13 b^2)(e^2 + 13f^2))
. It can be seen that is the only possible value.
Hence 
is the only common factor.
