# degree of splitting field of palindromic polynomial

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?