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Let $p(x) \in \mathbb Q[x]$ be a palindromic polynomial of even degree 2n.

Let $K$ be the splitting field of $p(x)$. Prove that $[K:\mathbb Q] \leq 2^nn!$.

I know that the palindromic polynomial of even degree 2n has a form $x^nq(x+\frac{1}{x})$ for some $q(x) \in \mathbb Q[x]$. Then it must be degree n.

With this fact, how can I solve the problem only with field extension and definition of splitting field?

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