What is the relationship between $[K:F]$ and the degree of the polynomial $f$?

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  • Thread starter Chris L T521
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In summary, the degree of a polynomial in a field extension represents the dimension of the extension field and directly corresponds to the degree of the extension. This relationship is known as the degree theorem and has implications in field theory. There is a limit to the degree of a polynomial, which is equal to the degree of the extension field. This relationship can also be applied to other mathematical concepts such as algebraic numbers and has applications in algebraic geometry and Galois theory.
  • #1
Chris L T521
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Here's this week's problem!

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Problem: Let $F$ be a field, $f\in F[x]$ be a polynomial of degree $n$, and let $K$ be a splitting field of $f$. Prove that $[K:F]$ divides $n!$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
No one answered this week's problem. You can find the solution below.

[sp]Since $K$ is a splitting field of $f\in F[x]$, $K/F$ is a Galois extension, and a Galois automorphism is determined by its action on the roots of $f$. This action can only permute the roots (since it must be an automorphism, and it must fix $f$); therefore, the Galois group $G$ is a subgroup of $S_n$ and thus $|G| = [K:F]$ divides $|S_n|=n!$.$\hspace{.25in}\blacksquare$[/sp]
 

FAQ: What is the relationship between $[K:F]$ and the degree of the polynomial $f$?

What is the significance of the degree of a polynomial in relation to its field extension?

The degree of a polynomial refers to the highest exponent present in the polynomial. In the context of a field extension, this degree represents the dimension of the extension field over the base field. In other words, it indicates the number of basis elements needed to span the extension field.

How does the degree of a polynomial affect the degree of its field extension?

The degree of a polynomial directly corresponds to the degree of its field extension. This means that a polynomial of degree n will result in a field extension of degree n. This relationship is known as the degree theorem.

Is there a limit to the degree of a polynomial when considering field extensions?

Yes, there is a limit to the degree of a polynomial in relation to its field extension. This limit is known as the degree bound and is equal to the degree of the extension field over the base field. In other words, the degree of the polynomial cannot exceed the degree of the field extension.

How does the relationship between $[K:F]$ and the degree of a polynomial impact field theory?

The relationship between the degree of a polynomial and the field extension has significant implications in field theory. It helps us understand the structure and properties of fields, as well as the behavior of polynomials in these fields. This relationship also allows us to classify fields into distinct categories based on their degree and other properties.

Can the relationship between $[K:F]$ and the degree of a polynomial be applied to other mathematical concepts?

Yes, the relationship between the degree of a polynomial and field extension can be applied to other mathematical concepts. For example, it is closely related to the concept of algebraic numbers, which are roots of polynomials with coefficients in a given field. This relationship also has applications in algebraic geometry and Galois theory.

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