Splitting field of a polynomial over a finite field

In summary, the conversation discusses the proof that a splitting field for an irreducible polynomial f over a field F of size p^r is F[x]/(f). The conversation mentions using Kronecker's theorem and the fact that finite extensions of finite fields are Galois, normal, and separable. It also suggests using the substitution homomorphism to show that the root of f is in F[x]/(f).
  • #1
resolvent1
24
0

Homework Statement


Assume F is a field of size p^r, with p prime, and assume [tex]f \in F[x][/tex] is an irreducible polynomial with degree n (with both r and n positive).

Show that a splitting field for f over F is [tex]F[x]/(f)[/tex].

Homework Equations


Not sure.

The Attempt at a Solution


I know from Kronecker's theorem that f has a root in some extension field of F, but I don't know that this root is necessarily in F[x]/(f). If I could obtain this, I could use the fact that finite extensions of finite fields are Galois, therefore normal (and separable), so f splits in F[x]/(f).
I also know that finite extensions of finite fields are simple, so [tex]F[x]/(f) \cong F(\alpha)[/tex] for some [tex]\alpha[/tex]. Then the substitution homomorphism ([tex]g \rightarrow g(\alpha)[/tex]) might help, if I knew that [tex]\alpha[/tex] is a root of f.

Thanks in advance.
 
Physics news on Phys.org
  • #2
I think you're thinking too hard. You want to find some polynomial expression in x that you can plug into f to get something that is divisible by f(x), right?
 
  • #3
I've got it, thanks.
 

FAQ: Splitting field of a polynomial over a finite field

What is a splitting field?

A splitting field of a polynomial over a finite field is the smallest field extension that contains all the roots of the polynomial. This means that every root of the polynomial can be expressed as a linear combination of elements in the splitting field.

Why is the concept of a splitting field important in finite fields?

In finite fields, not all polynomials have roots. Therefore, the splitting field allows us to extend the finite field to include the roots of a given polynomial, making it possible to solve equations and perform other algebraic operations.

How is the splitting field of a polynomial over a finite field determined?

The splitting field can be constructed by adjoining all the roots of the given polynomial to the base finite field. This process can be repeated until all the roots have been added, resulting in the smallest field extension that contains all the roots.

What are the properties of a splitting field?

A splitting field is a finite field, meaning it has a finite number of elements. It is also a Galois extension, which means it is a field extension that is invariant under all automorphisms. Additionally, the degree of the splitting field is equal to the degree of the polynomial.

Can a splitting field have multiple polynomials?

Yes, a splitting field can have multiple polynomials. This is because a splitting field is not unique and can be constructed for any polynomial over a finite field. However, the degree of the splitting field will be the same for all polynomials that have the same roots.

Similar threads

Back
Top