Galois Theory - Fixed Subfield of K by H ....

[Math Amateur]

I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ...

I am currently focused on Chapter 8: Galois Theory, Section 1: Automorphisms of Field Extensions ... ...

I need help with Proposition 11.1.11 on page 560 ... ...Proposition 11.1.11 reads as follows:

In the above Proposition from Lovett we read the following:" ... ... Since $\sigma$ is a homomorphism $U(F) \ \rightarrow \ U(F)$ ... ... "My question is ... ... what is $F$ ... is it a typo ... should it be $K$...Hoping someone can help ... ...

PeterNOTE: $U(F)$ in Lovett means the group of units of the ring $F$ ...

 

[andrewkirk]

It looks like a typo to me. $F$ is first mentioned at the end of the second line, in $(F,+)$. If we replace $F$ by $K$ the proof still works.

Is it possible that, in the context of the book, $K$ has been assumed to be a sub-field of some larger field $F$? If so, and that has been stated in the preceding pages, then it wouldn't be a typo.

It's also not clear to me what $U(F)$ means, although from the context I'm guessing it's referring to the multiplicative group $(F-\{0\},\times)$ which, if we assume that $F$ really means $K$, is $(K-\{0\},\times)$.

 

[fresh_42]

FWIW. I recall a situation, in which $F \subseteq K$ has been the extension. I remember it, as I confused both and finally built a mnemonic $F =$ fixed elements.

 

[Math Amateur]

FWIW. I recall a situation, in which $F \subseteq K$ has been the extension. I remember it, as I confused both and finally built a mnemonic $F =$ fixed elements.

Hmm ... yes ... Lovett often uses K/F for a field extension ... but still ... seems to me that that doesn't resolve the problem of the exact nature of F ...

Do you think that Lovett meant K when he wrote F?

Peter

 

[fresh_42]

I have the same difficulties as @andrewkirk to understand what $F$ and $U(F)$ are. $U(.)$ could be units, the multiplicative group of a field, and $F=K$ which makes sense, if I didn't miss something. As the entire topic is about the correspondence between fields and automorhism groups, it might well be that a $F$ somehow found its way into the proof although it should have been $K$.

 

[Math Amateur]

HI fresh_42 ...

The notation U(K) in Lovett is the group of units of K ...

Thanks ... I also think F should be K ... glad to have your agreement ...

Peter