Let $E$ be an extension field of $F$, of dimension $n$. Is there an element of degree $n$ in $E$ over $F$?

Question

In other words, does there exist any element in $E$ such that its minimal polynomial over $E$ is of degree $n$?

Answer 1

What you are asking is whether any finite field extension has a primitive element; that is, if $[E:F]=n$, does there exist an $a\in E$ such that $E=F(a)$.

To see that this is equivalent to what you are asking, note that if such an element exists, then its minimal polynomial must have degree $[F(a):F] = [E:F] = n$; conversely, if there is an $a$ with minimal polynomial of degree $n$, then $[E:F(a)][F(a):F] = [E:F]=n$, but $[F(a):F]=n$, and therefore $[E:F(a)]=1$ so $E=F(a)$.

Such an extension is called a simple extension.

The answer is that such an element does not always exist, but it exists in most standard situations.

The most common situation is separability.

Primitive Element Theorem If $E$ is finite and separable over $F$, then $E/F$ is simple.

More generally, we have:

Theorem. Let $E$ be a finite extension of $F$. Then the following are equivalent:

1. The extension $E/F$ is simple.
2. There are only finitely many intermediate fields $L$, $F\subseteq L\subseteq E$.

You can find proofs of this in this site, for example here.

You can also find examples of finite extensions where it does not happen; they have to be inseparable, which means they have to be in characteristic $p$ and infinite fields. You can an example here