Let $L$ be the subfield of $\mathbb{R}$, of all reals that are algebraic over $\mathbb{Q}$: $L = \{ x\in \mathbb{R} : x \text{ is algebraic over } \mathbb{Q} \}, \;\;\; \mathbb{Q} \subseteq L$.
Let $K$ be finite field extension of $L$ such that $K \subseteq \mathbb{C}$. By the primitive element theorem $K = L(a)$. Denote by $f(x)$ the minimal polynomial of $a$ over $L$.
Prove $f(x)$ has no real roots.
Any hints ?
Edit: I made a mistake in the original question. $K/L$ is finite extension.