TITLE: A local-global problem concerning roots of polynomials
QUESTION [12 upvotes]: Let $f(x)$ be a polynomial with integer coefficients, irreducible over the integers. Suppose that for all primes $p$, $f$ has a zero in the field $\mathbb{Q}_p(\sqrt{2})$. Here $\mathbb{Q}_p$ denotes the field of $p$-adic numbers. Must $f$ have a zero in the field $\mathbb{Q}(\sqrt{2})$?

REPLY [3 votes]: Peter Mueller, answering my question here, points me to "On a question of W. Jehne concerning covering subgroups of groups and Kronecker classes of fields" by Jan Saxl. On the first page, we see 

Corollary: Let $K = \mathbb{Q}(\sqrt{d})$ be a quadratic extension of $\mathbb{Q}$ with $d$ a square free integer and let $M$ be a proper finite extension of $K$. Then there are infinitely many prime numbers $p$ with $\left( \frac{d}{p} \right)=1$ which have no prime divisor in $M$ of first degree.

In other words, for any $f$ irreducible over $\mathbb{Q}$ which does not split completely over $K$, there are infinitely many $p$ for which $p = \pi \bar{\pi}$ in $K$, with $K_{\pi} \cong \mathbb{Q}_p$ and $f$ having no roots in $K_{\pi}$.