Galois Extension: Proving L is Galois Over K

  • Thread starter math_grl
  • Start date
  • Tags
    Extension
In summary, the conversation discusses the extension of a field automorphism from a Galois extension M over K to a Galois extension L over K. It is mentioned that L is Galois over M and M is Galois over K. The speaker asks for help in understanding why L is also Galois over K. The response includes a summary of the steps involved in proving L is indeed Galois over K, including the use of automorphisms and theorems from Dummit and Foote's book.
  • #1
math_grl
49
0
This stuff is killing me...

Let [tex]K \leq M \leq L[/tex] be fields such that L is galois over M and M is galois over K. We can extend [tex]\phi \in G(M/K)[/tex] to an automorphism of L to show L is galois over K.

I need help filling in the details in why exactly L is galois over K.
 
Physics news on Phys.org
  • #2
Something like this, I think: let [tex][M:K]=a[/tex] and [tex][L:M]=b[/tex]. Since [tex]M/K[/tex] is Galois, there are [tex]a[/tex] automorphisms of [tex]M[/tex] that fix [tex]K[/tex]. For the same reason, there are [tex]b[/tex] automorphisms of [tex]L[/tex] that fix [tex]M[/tex]. So there are [tex]ad[/tex] automorphisms of [tex]L[/tex] that fix [tex]K[/tex]. Since [tex]ad[/tex] is also the degree of [tex]L/K[/tex], it's Galois.

If you're using Dummit and Foote, check out theorems 13.8 and 13.27.
 

1. What is a Galois extension?

A Galois extension is a type of field extension in abstract algebra that is used to study the properties of polynomial equations. It is named after the French mathematician Évariste Galois, who first developed the theory in the 19th century. In a Galois extension, the field extension is closed under a group of automorphisms, which allows for a deeper understanding of the structure and solutions of polynomial equations.

2. How is a Galois extension proven to be Galois over K?

To prove that a field extension L over a field K is Galois, we need to show that two conditions are satisfied: (1) L is a splitting field of a polynomial over K, and (2) L is a normal extension of K. This means that all the roots of the polynomial are contained in L and that L is closed under the automorphisms of K. If both of these conditions are met, then L is considered to be a Galois extension over K.

3. What is the significance of a Galois extension?

Galois extensions are important because they allow us to understand the structure of polynomial equations and their solutions. The theory of Galois extensions has applications in many areas of mathematics, including algebraic geometry and number theory. It also has practical applications in fields such as cryptography, where understanding the properties of polynomial equations is crucial.

4. How are automorphisms related to Galois extensions?

Automorphisms play a key role in Galois extensions because they are the transformations that preserve the structure of the field. In a Galois extension, the automorphisms of the base field K are also automorphisms of the extension field L. This means that the solutions to a polynomial equation in L will remain valid under any of the automorphisms of K. This allows for a deeper understanding of the properties and solutions of polynomial equations.

5. Are all Galois extensions finite?

No, not all Galois extensions are finite. While many examples of Galois extensions are finite, such as Galois fields, there are also infinite Galois extensions. An example of an infinite Galois extension is the field of rational functions, which is a Galois extension of the field of rational numbers. However, the Galois group of an infinite Galois extension will also be infinite, making the study of infinite Galois extensions more challenging.

Similar threads

  • Linear and Abstract Algebra
Replies
8
Views
979
  • Linear and Abstract Algebra
Replies
6
Views
1K
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
856
  • Linear and Abstract Algebra
Replies
5
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
921
Back
Top