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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.

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