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

[Math Amateur]

I am reading Dummit and Foote, Chapter 14 - Galois Theory.

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 ... ... :

 

[jambaugh]

If you consider the group of automorphisms of K that fix F, that group may in fact fix more than just F, namely F1 making F1 the fixed field.

I'm very rusty on my Galois Theory but this is true for Lie groups too when you consider automorphisms of a Lie group vs inner automorphisms.

 

[Math Amateur]

Hi jambaugh ... thanks for the help ...

OK can see that ... that seems to explain it ...

Thanks again,

Peter