# Prime ideal in ring

1. May 8, 2007

### ElDavidas

1. The problem statement, all variables and given/known data

Take the ideal

$$I = < 6, 3 + 3 \sqrt{-17} >$$

in the ring $Z [ \sqrt{-17} ]$.

Determine whether this ideal is prime or not.

2. Relevant equations

$$<18> = I^2$$

There is no element $\alpha \in Z [ \sqrt{-17} ]$ such that $18 = \alpha^2$

3. The attempt at a solution

I really don't know how to go about doing this. I have the definition of a prime ideal P.

$P$ is a prime ideal $\Leftrightarrow$ if $ab \in P$ then $a \in P$ or $b \in P$.

And I see that $< 6, 3 + 3 \sqrt{-17} > = 6 Z [ \sqrt{-17} ] +( 3 + 3 \sqrt{-17}) Z [ \sqrt{-17} ]$.

Is it also possible to pull out the $Z [ \sqrt{-17} ]$ from the above equation?

Thanks

Last edited: May 8, 2007