Possible Duplicate:
Are quartic minimal polynomials over $\mathbb{Q}$ always reducible over $\mathbb{F}_p$?
This question was originally a homework problem for an algebra course, but the professor took it off the homework once he realized it was too hard. However, I am curious. Here is the question:
If $K = \mathbb{Q}(\theta)$ is a biquadratic extension of $\mathbb{Q}$ and the minimal polynomial $f(x)$ of $\theta$ over $\mathbb{Q}$ has coefficients in $\mathbb{Z}$, prove that $f(x)$ irreducible over $\mathbb{Q}$ but that $f(x)$ mod $p$ is reducible over $\mathbb{F}_p$ for every prime number $p$.
Give an explicit example of such a polynomial $f(x)$
If anyone could even give me an example, that would be wonderful.
