degree of splitting field of palindromic polynomial

Asked Oct 03 '18 at 15:25

Active Oct 03 '18 at 15:25

Viewed 103 times

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?

edited Jun 12 '20 at 10:38

Community

asked Oct 03 '18 at 15:25

Pearl

1

Use the same idea as in https://math.stackexchange.com/questions/2940360/degree-of-splitting-field-of-pqx – Angina Seng Oct 03 '18 at 15:26
1

See here for this specific case, and Lord Shark's link for a nice generalization. – Jyrki Lahtonen Oct 03 '18 at 16:04
Thanks for advice! – Pearl Oct 05 '18 at 00:19

degree of splitting field of palindromic polynomial

0 Answers0