Prime ideal question (abstract algebra)

  • #1

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?
 

Answers and Replies

  • #2
22,129
3,297
Hi Metric_Space! :smile:

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

The first thing I would do is calculate [itex]\mathbb{Z}[\sqrt{10}]/\mathfrak{p}[/itex] and check whether this is an integral domain...
 
  • #3
Right..and there is a theorem that relates Z[10−−√]/p to prime ideals I think, right?
 
  • #4
22,129
3,297
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...
 
  • #5
cool..yes, I have a theorem like the one you mention -- that's a good start. Thanks for the hint.
 

Related Threads on Prime ideal question (abstract algebra)

  • Last Post
Replies
0
Views
885
  • Last Post
Replies
3
Views
6K
Replies
3
Views
792
Replies
6
Views
4K
Replies
5
Views
3K
Replies
3
Views
531
Replies
3
Views
2K
Replies
2
Views
2K
Replies
0
Views
944
Replies
5
Views
1K
Top