Field extensions and Galois goup

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In summary: From the previous we also know that is Galois. From the theorem we also know that is cyclic and is generated by the automorphism . We can calculate the order of by calculating the smallest integer such that . So we conclude that .
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mathmari
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Hey! :eek:

Let b a prime, and let be irreducible. Let be a root pf .

We have that , where is a basis of , so . There are choices for each . So, there are choices for . Therefore, we have that , right? (Wondering)

I want to show that contains all the roots of .

I holds that every finite extension of a finite field is normal, right? But how can we prove this? (Wondering)

Then I have shown that for each , is a root of . I have also shown that and for some .

Then I want to show that is cyclic and let be a generator. I want to calculate also the order of as a function of .

In the book there is the following corollary:
View attachment 6352

We have that and . From the above we have that is Galois, , which is true since . From the theorem we also have that is cyclic and is generated by the automorphism .

How can we calculate the order of ? (Wondering)

Then independent from that I want to calculate a simple expression of .

I have done the following:
Since is a generator of that means that is a -automorphism of that maps to a root of to an other root, right? (Wondering)

So, we have that . Therefore, we get the following:
Is this correct? (Wondering)
The order of is the smallest integer such that . Does this imply that ? (Wondering)
 

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There is a typo at . It should be .
 
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mathmari said:
Let b a prime, and let be irreducible. Let be a root pf .

We have that , where is a basis of , so . There are choices for each . So, there are choices for . Therefore, we have that , right? (Wondering)

I want to show that contains all the roots of .
To show that we have to show that the extension is normal, right? (Wondering)

Since we get that for each it holds that .
So, each element of is a root of .
From that we don't get yet that is the spitting field of , do we? (Wondering)
 

FAQ: Field extensions and Galois goup

What is a field extension?

A field extension is a mathematical concept that involves extending a base field by adjoining new elements to it. This allows for the creation of a larger field that contains the base field and the newly added elements.

What is a Galois group?

A Galois group is a group that represents the symmetries of a field extension in algebra. It is named after the French mathematician Évariste Galois and provides important information about the structure of a field extension.

How are field extensions and Galois groups related?

Field extensions and Galois groups are closely related, as the Galois group is used to study the properties and structure of a field extension. In particular, the Galois group helps determine whether a field extension is solvable by radicals.

What is the significance of Galois groups in mathematics?

Galois groups have significant applications in various areas of mathematics, including algebra, number theory, and geometry. They provide a powerful tool for understanding and solving problems related to field extensions and their properties.

How do you determine the order of a Galois group?

The order of a Galois group is equal to the degree of the field extension. This means that if the field extension has a degree of n, then the Galois group will have n elements. This relationship is known as the fundamental theorem of Galois theory.

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