Maximal ideals of $\mathbb{Z}[X]$ are of the form $(p, f(X))$, where $f$ reduces to an irreducible polynomial mod $p$. Now, given a principal prime ideal $\mathfrak{p}$ of $\mathbb{Z}[X]$, I would like to extend it to a maximal ideal.

If $\mathfrak{p}=(p)$ for some prime number $p$, this is easy. Just adjoin a polynomial which is irreducible mod $p$. In the other case, when $\mathfrak{p}=(f)$ for some $f$ irreducible in $\mathbb{Q}[X]$, I am having some trouble.

If $f$ is also irreducible modulo some prime, we can simply adjoin that prime to get a maximal ideal. But my problem lies in polynomials which are irreducible in $\mathbb{Q}[X]$ but reducible modulo every prime. A simple example is $X^4+1$.

How do I extend such primes to maximal ideals?


You could pick a prime number $p$, and one irreducible factor of $f$ modulo $p$.

  • $\begingroup$ Ah, I was looking from a strange angle, thought that didn't work. Thanks! $\endgroup$ – maarten Nov 6 '16 at 16:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.