I am reading Dummit and Foote, Chapter 14 - Galois Theory.(adsbygoogle = window.adsbygoogle || []).push({});

I am currently studying Section 14.2 : The Fundamental Theorem of Galois Theory ... ...

I need some help with Corollary 10 of Section 14.2 ... ... and the definition of ##\text{Aut}(K/F)## ... ...

Corollary 10 reads as follows:

Now the Definition of ##\text{Aut}(K/F)## is as follows:

Now in Corollary 10 we read the following:

" ... ... Then

## | \text{Aut}(K/F) | \ \le \ [ K \ : \ F ]##

with equality if and only if ##F## is the fixed field of ##\text{Aut}(K/F)## ... ... "

My question is as follows:

Given the definition of ##\text{Aut}(K/F)## shown above, isn't ##F## guaranteed to be the fixed field of ##\text{Aut}(K/F)## ... ... ?

Hope someone can resolve this problem/issue ...

Help will be much appreciated ...

Peter

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The above post will be easier to follow if readers understand D&F's definition of a Galois Extension and a Galois Group ... so I am providing the definition as follows ... ... :

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# I Galois Theory - Fixed Field of F and Definition of Aut(K/F)

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