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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:
View attachment 6666
Now the Definition of $$\text{Aut}(K/F)$$ is as follows:
https://www.physicsforums.com/attachments/6667
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===========================================================================================
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 ... ... :
https://www.physicsforums.com/attachments/6668
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:
View attachment 6666
Now the Definition of $$\text{Aut}(K/F)$$ is as follows:
https://www.physicsforums.com/attachments/6667
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===========================================================================================
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 ... ... :
https://www.physicsforums.com/attachments/6668