Prime ideal question (abstract algebra)

[Metric_Space]

Homework Statement

Let D = Z[sqrt(10)], and let P be the ideal (2,sqrt(10)) 10). Prove that P is a prime
ideal of D.

Homework Equations

The Attempt at a Solution

Not sure where to start. I think elements are of the for a+b*sqrt(10), a,b integers.

Any hints as to what to do next?

 

[micromass]

Hi Metric_Space!

So your ideal is \mathfrak{p}=(2,\sqrt{10})?

The first thing I would do is calculate \mathbb{Z}[\sqrt{10}]/\mathfrak{p} and check whether this is an integral domain...

 

[Metric_Space]

Right..and there is a theorem that relates Z[10−−√]/p to prime ideals I think, right?

 

[micromass]

Right..and there is a theorem that relates Z[10−−√]/p to prime ideals I think, right?

Uuh, maybe. But what I'm going for is that p is a prime ideal in a commutative ring R if and only if R/p is an integral domain...

 

[Metric_Space]

cool..yes, I have a theorem like the one you mention -- that's a good start. Thanks for the hint.