automorphism

**heathrowjohnny** · March 1st 2008, 06:24 PM

If $\sigma_{1}, \sigma_2, \ldots, \sigma_{n}$ is a group of automorphisms of a field $E$ and if $F$ is a fixed field of $\sigma_{1}, \sigma_{2}, \ldots, \sigma_{n},$ then $(E/F) = n$ .

How would I prove this?

**ThePerfectHacker** · March 1st 2008, 07:11 PM

Originally Posted by heathrowjohnny

If $\sigma_{1}, \sigma_2, \ldots, \sigma_{n}$ is a group of automorphisms of a field $E$ and if $F$ is a fixed field of $\sigma_{1}, \sigma_{2}, \ldots, \sigma_{n},$ then $(E/F) = n$ .

There is a result due to Artin

. Let $G$ be a finite group of automorphism of $E$ , let $F=E^G$ be the fixed subfield, then $E/F$ is finite and $[E:F] \leq |G|$ . Note, you need the $\leq$ sign.

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