Let $F$ be an algebraic Number Field and $O_F$ its ring of integers. If $\alpha \in O_F$ then is there a criterion for when $\alpha$, \alpha' are relatively prime?\ \ I am most interested in $O_F=\mathbb{Z}[\zeta_3]$ for the time being.\ \ It seems like a possible criterion could be Let $a,b\in \mathbb{Z}$ and $\alpha=a+b\zeta$. Then, $gcd(a,b)=1$ iff \alpha, \alpha' are relatively prime. Is this correct? The $\Rightarrow$ direction is most important to me.\ \ Thanks.\ \
Is there a criterion for when an algebraic integer and its conjugate are relatively prime?
1
$\begingroup$
number-theory
-
0Yes that is true as well. Thanks. – 2011-02-28
1 Answers
2
In your case we have $\gcd(a + b \zeta, a + b \zeta^2) = \gcd(a + b \zeta, b(\zeta - 1))$. If $\gcd(a, b) = 1$ then this equals $\gcd(a + b \zeta, \zeta - 1) = \gcd(a + b, \zeta - 1)$ which equals $1$ if and only if $3 \nmid a + b$.
In general I am not sure what you mean by "relatively prime" if $\mathcal{O}_F$ is not a UFD. Do you mean that they generate the unit ideal? There are lots of ways to check this generalizing the method above. A sufficient condition is that (\alpha, \alpha') = (x, y) where the norms of $x, y$ are relatively prime.