Try to work with an ideal generated by two elements.
Hi everyone, this is my first post. Hope you don't mind that I'm using it to ask a homework question.
Here's the problem:
- Show is a prime ideal.
- Show is not maximal.
For 1), I let and ; that is, . Because is an integral domain, it has no zero divisors; thus, either or must be and hence in . Is this right?
I wasn't sure about 2); can I take or ?
Thanks for your help clic-clac. I'm still a little confused, though.
The polynomials such that f(0) = 0 are all polynomials with constant term 0. So I need to find an ideal that contains these and a polynomial with a non-zero constant term. So could I take ?
Your new answer isn't so far from a solution: the idea of "adding" , which wasn't in your first ideal, is good.
But a reunion of ideals is, in general, not an ideal. The construction that transforms a reunion of ideals into an ideal is the ideal generated by a set.
To simplify notations, do you agree that is the ideal in ?
Since is not a ideal, we can consider which is also written or
contains and so it strictly contains . The question now is
So take a i.e. Is possible? (That will answer the problem, since for any ideal )