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

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Chris L T521

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Jan 26, 2012
995
Here's this week's problem!

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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 $n!$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Chris L T521

Well-known member
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Jan 26, 2012
995
No one answered this week's problem. You can find the solution below.

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 $|S_n|=n!$.$\hspace{.25in}\blacksquare$
 
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