Splitting Fields - Example 3 - D&F Section 13.4, pages 537 -

In summary, Dummit and Foote explain that the degree of a proper extension of a field must be at least two, and that if no further vector is needed in a basis for the extension, then the extension is identical to the original field.
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I am reading Dummit and Foote, Chapter 13 - Field Theory.

I am currently studying Section 13.4 : Splitting Fields and Algebraic Closures ... ...

I need some help with an aspect of Example 3 of Section 13.4 ... ...

Example 3 reads as follows:
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In the above text by Dummit and Foote, we read the following:

" ... ... Since ##\sqrt{ -3 }## satisfies the equation ##x^2 + 3 = 0## the degree of this extension over ##\mathbb{Q} ( \sqrt [3] {2} )## is at most ##2##, hence must be ##2## since we observed above that ##\mathbb{Q} ( \sqrt [3] {2} )## is not the splitting field ... ... "I do not understand why the degree of the extension ##K## over ##\mathbb{Q} ( \sqrt [3] {2} )## must be exactly ##2## ... ... why does ##\mathbb{Q} ( \sqrt [3] {2} )## not being the splitting field ensure this ... ...

Can someone please give a simple and complete explanation ...

Hope someone can help ...

Peter
 

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The degree of any proper extension of a field must be at least two because the degree is the dimension of the extension considered as a vector space over the smaller field.

Given any nonzero vector in a finite-dimensional vector space, there exists a basis containing that vector. So we can take the 1 of the smaller field as the first vector in a basis for the extension. If no further vector is needed in the basis then the extension is identical to the original field. Hence the degree must be at least two for the extension to be proper.

Since it must also be no greater than two, it must be two.
 
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andrewkirk said:
The degree of any proper extension of a field must be at least two because the degree is the dimension of the extension considered as a vector space over the smaller field.

Given any nonzero vector in a finite-dimensional vector space, there exists a basis containing that vector. So we can take the 1 of the smaller field as the first vector in a basis for the extension. If no further vector is needed in the basis then the extension is identical to the original field. Hence the degree must be at least two for the extension to be proper.

Since it must also be no greater than two, it must be two.
Thanks Andrew ... just now reflecting on what you have said ...

Peter
 

FAQ: Splitting Fields - Example 3 - D&F Section 13.4, pages 537 -

1. What is a splitting field?

A splitting field is a field extension that contains all the roots of a given polynomial over a smaller field. It is the smallest field extension that allows the polynomial to be factored completely into linear factors.

2. How is a splitting field related to a polynomial?

A splitting field is related to a polynomial as it is the field extension in which the polynomial completely factors into linear factors. In other words, the roots of the polynomial can be found in the splitting field.

3. What is the degree of a splitting field?

The degree of a splitting field is the degree of the polynomial it is splitting. This means that if the polynomial has degree n, the splitting field will have degree n over the base field.

4. How do you construct a splitting field?

To construct a splitting field, you start with the base field and adjoin all the roots of the polynomial. This will create a field extension that contains all the roots of the polynomial and is the smallest field extension that allows for this.

5. Why are splitting fields important in algebra?

Splitting fields are important in algebra because they allow us to fully factor polynomials into linear factors. This is useful in many areas of mathematics, including number theory and algebraic geometry. Splitting fields also have applications in algebraic coding theory and cryptography.

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