Showing a finite field has an extension of degree n

[ttzhou]

Homework Statement

Suppose $F$ is a finite field and $n > 0$.

Show that $F$ has a field extension of degree $n$

Homework Equations

Tower Law.

The Attempt at a Solution

Let $p^m$ be the size of $F$

It's trivial to note by characterization that there exists a finite field $G'$ of size $p^{mn}$ which has a subfield $G$ of size $p^m$.

Then $G \cong F$; by Tower Law, $[G' : G]$ is of degree n.

I'm having trouble showing that there is an extension of $F$ that has degree n, however; I tried finding some isomorphism and using theorems about algebraic elements, but for some reason this stuff is not clicking in my head.

I'm beginning to think the question only requires that I show that there is a field extension of an isomorphic copy, but that can't be right.

Please give tiny hints only that can push me in the right direction or clarify some glaring error I've made.

Thanks for your time.

 

[mighty2000]

Hi there,

You are on the right track! The key to this problem is to use the fact that F is a finite field, which means it has a finite number of elements. Since we know that there exists a finite field G' of size p^{mn} with a subfield G of size p^m, we can use the fact that G \cong F to show that there must be a field extension of F with degree n.

To do this, consider the elements of G' that are not in G. These elements must be algebraic over G, since G is a finite field and therefore every element of G' must be algebraic over it. This means that there is some polynomial of degree n that has these elements as roots. Therefore, we can construct a field extension of F by adjoining these elements to F and considering the field generated by F and these elements. This field extension will have degree n, since the polynomial we used to construct it has degree n.

I hope this helps! Let me know if you need any further clarification.