Problem of the Week #92 - March 3rd, 2014

[Chris L T521]

Here's this week's problem!

-----

Problem: Let $F$ be a field, $f\in F[x]$ be a polynomial of degree $n$, and let $K$ be a splitting field of $f$. Prove that $[K:F]$ divides $n!$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

 

[Chris L T521]

No one answered this week's problem. You can find the solution below.

Spoiler

Since $K$ is a splitting field of $f\in F[x]$, $K/F$ is a Galois extension, and a Galois automorphism is determined by its action on the roots of $f$. This action can only permute the roots (since it must be an automorphism, and it must fix $f$); therefore, the Galois group $G$ is a subgroup of $S_n$ and thus $|G| = [K:F]$ divides $|S_n|=n!$.$\hspace{.25in}\blacksquare$