Galois extension, lattice of subfields

In summary, K=\mathbb{Q}(\sqrt{2+\sqrt{2}}) is a Galois extension of \mathbb{Q} with \text{Gal}(K/\mathbb{Q})=\mathbb{Z}_4. The lattice of subfields can be determined by looking at the lattice of subgroups of the Galois group. This can help to better visualize and understand the subfields within K.
  • #1
mathsss2
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[tex]K=\mathbb{Q}(\sqrt{2+\sqrt{2}})[/tex] is a Galois extension of [tex]\mathbb{Q}[/tex] [I showed this]. Determine [tex]\text{Gal}(K/\mathbb{Q})[/tex] and describe the lattice of subfields [tex]\mathbb{Q} \subset F \subset K[/tex].

I found that [tex]\text{Gal}(K/\mathbb{Q})=\mathbb{Z}_4[/tex]. I do not know how to draw the lattice of subfields, so far I have this: [see attachment], but I do not think it is right. I need help with the lattice of subfields.
 

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  • #2
Recall that the lattice of subfields is directly correlated to the lattice of subgroups of the galois group. This should make it easier
 

1. What is a Galois extension?

A Galois extension, named after mathematician Évariste Galois, is a field extension that has a certain structure and properties. It is a finite field extension in which the intermediate fields form a lattice. This means that every intermediate field between the base field and the extension field can be reached by a sequence of simple extensions.

2. What is a lattice of subfields?

A lattice of subfields is a mathematical structure that represents the relationships between the intermediate fields in a Galois extension. It consists of a partially ordered set of subfields, where each subfield is related to its immediate subfields by inclusion. This lattice structure is important in understanding the properties of a Galois extension, such as the degree of the extension and the automorphisms of the extension.

3. What is the significance of a Galois extension?

Galois extensions are significant because they have many properties that make them useful in algebra and number theory. They allow us to study the roots of polynomials and solve equations, and they also have connections to other mathematical structures such as groups and fields. In particular, the Galois group of a Galois extension is a fundamental group that encodes important information about the extension.

4. How do you determine the lattice of subfields in a Galois extension?

The lattice of subfields in a Galois extension can be determined by finding all intermediate fields between the base field and the extension field. This can be done by finding the roots of the minimal polynomial of the extension and then constructing the subfields using these roots. The lattice can also be determined by examining the automorphisms of the Galois extension and their fixed fields.

5. What are some applications of Galois extensions?

Galois extensions have many applications in mathematics and other fields. In algebra, they are used to solve polynomial equations and study group theory. In number theory, they are used to study algebraic number fields and their properties. They also have applications in cryptography, coding theory, and physics. Galois extensions are a fundamental concept in abstract algebra and have many diverse applications in various branches of mathematics.

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