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

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In summary, Proposition 11.1.11 in Chapter 8, Section 1 of "Abstract Algebra: Structures and Applications" by Stephen Lovett discusses the homomorphism U(K) \rightarrow U(K) and a question arises about the variable F being a typo and if it should actually be K. This is confirmed by another user, Euge, and the summary concludes with Peter thanking them for their help.
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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:
https://www.physicsforums.com/attachments/6664
In the above Proposition from Lovett we read the following:" ... ... Since \(\displaystyle \sigma\) is a homomorphism \(\displaystyle U(F) \ \rightarrow \ U(F)\) ... ... "My question is ... ... what is \(\displaystyle F\) ... is it a typo ... does it mean \(\displaystyle K\) ...Hoping someone can help ... ...

PeterNOTE: U(F) in Lovett means the group of units of the ring F ...
 
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You are correct, $F$ should be $K$.
 
  • #3
Euge said:
You are correct, $F$ should be $K$.
Thanks Euge ...

Peter
 

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

What is Galois Theory?

Galois Theory is a branch of abstract algebra that studies the relationship between field extensions and their corresponding symmetry groups. It was developed by French mathematician Évariste Galois in the 19th century.

What is a fixed subfield in Galois Theory?

A fixed subfield, also known as a fixed field, is a subset of a larger field that remains unchanged under the action of a given subgroup. In Galois Theory, fixed subfields are important for understanding the structure and properties of field extensions.

How is Galois Theory applied in mathematics?

Galois Theory has many applications in mathematics, particularly in the study of field extensions and their corresponding symmetry groups. It is also used in other branches of mathematics, such as number theory and algebraic geometry.

What is the significance of the Galois group in Galois Theory?

The Galois group is a key concept in Galois Theory, representing the symmetry group of a given field extension. It provides important information about the structure and properties of the extension, and can help determine whether a given polynomial equation is solvable by radicals.

How does Galois Theory relate to other branches of mathematics?

Galois Theory has connections to many other areas of mathematics, including group theory, algebraic geometry, and number theory. It provides a powerful framework for studying the properties of field extensions, which have applications in a wide range of mathematical fields.

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