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

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In summary, Peter is having difficulty understanding what Proposition 11.1.11 from Abstract Algebra: Structures and Applications is saying. He is hoping someone can help him.
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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:
?temp_hash=b73d2c3295f7984285d98c67b26cdd93.png

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## ...
 

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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)##.
 
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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.
 
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  • #4
fresh_42 said:
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
 
  • #5
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##.
 
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  • #6
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
 

1. What is Galois Theory?

Galois Theory is a branch of mathematics that studies the properties and structure of field extensions, which are algebraic structures that extend the operations of addition and multiplication from a smaller field to a larger field. It was developed by French mathematician Évariste Galois in the 19th century.

2. What is a fixed subfield in Galois Theory?

A fixed subfield, also known as a fixed field, is a subfield of a larger field that remains unchanged when subjected to a group of automorphisms. In Galois Theory, it refers to the subfield of a Galois extension that is fixed by the automorphisms of the Galois group.

3. What is K by H in Galois Theory?

In Galois Theory, K by H refers to the fixed subfield of a field extension K by a subgroup H of the Galois group. This means that the elements of K are left unchanged by any automorphism in the subgroup H.

4. What is the significance of fixed subfields in Galois Theory?

Fixed subfields play a crucial role in Galois Theory as they provide a way to study the properties of field extensions. By understanding the fixed subfields, we can determine the structure and solvability of polynomials, as well as the possibility of finding solutions to equations using radical expressions.

5. How is Galois Theory applied in other areas of mathematics?

Galois Theory has applications in many areas of mathematics, including algebraic geometry, number theory, and topology. It also has practical applications in cryptography and coding theory, as well as in physics and chemistry for studying symmetry and symmetry breaking in physical systems.

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