Check whether roots exist for a polynomial is splitting field of another polynomial

Question

Given that $F$ is the splitting field over $\mathbb{Q}$ of $x^{13}-1$.

If I have another polynomial, say $x^9-3x^5+7x+10$ , how can I say whether it will have roots in $F$ or not?

I know that for $x^{13}-1$, $F$ will be equal to $\mathbb{Q}\left(1, \cos\frac{2\pi}{13} +i\sin\frac{2\pi}{13}\right)$ but don't know how to proceed further.

Answer 1

Extending on the hint in comments. Say $\alpha$ is a root of the given polynomial $f$ of degree $9$.

$F(\alpha)\cong \frac{F[X]}{\langle f(x)\rangle}$

Then $[F(\alpha):F]=9$.

if $\alpha\in E$ where $E$ is the splitting field of polynomial $\sum_{i=0}^{12}x^{i}$ .

Then $F(\alpha)$ is a subfield of $E$ .

But by Tower Law $[E:F]=[E:F(\alpha)]\cdot[F(\alpha):F]\implies 12 = k\cdot 9$ for some integer $k$.

This is not possible.