Let $f \in \mathbb{Z}[x]$ be an irreducible monic polynomial, let $N$ be its splitting field, and let $G$ be the Galois group of the extension $N/\mathbb{Q}$. Let $p$ be a prime dividing the discriminant of $N$.

I would like to say that $f$ mod $p$ must admit a quadratic factor.

We know that $p$ ramifies in $N$ (because it divides the discriminant of $N$) and hence, provided $p$ is coprime to the conductor of $\mathbb{Z}[\theta]$ (where $\theta$ is a root of $f$), we may apply Dedekind's Theorem to obtain the result.

However, if $p$ does divide the conductor, then I am unsure about what to do.


Your Answer

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

Browse other questions tagged or ask your own question.