I am trying to prove the following statement.
Let $f(x)\in \mathbb{Z}[x]$ be a quartic polynomial with Galois group $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. Show that $f(x)$ is reducible modulo every prime $p>3$.
I don't know how to approach. It seems $f(x)$ need not to be even irreducible over $\mathbb{Q}$. I will appreciate any help! Thanks in advance!
Let $f\in \mathbb{Z}[x]$ be a polynomial which, when reduced mod $p$, has an irreducible factor of degree $n$. Then the Galois group of $\mathbb{Q}[x]/(f(x))$ has an $n$ cycle. For instance, you can see this in Lang's algebra, page 274. Since the Klein four group only has elements of order $2$, this should get what you want.