If $K:F$ is a finite extension and $F\subseteq E \subseteq K$, is $K:E$ also finite?

Question

Suppose that $F:E:K$ are field extensions and $K:F$ is finite. Then is $K:E$ necessarily finite?

Since $K:F$ is finite,$K$ has a basis $\{k_1,...,k_n\}$ over $F$. Indeed this set still spans $K$ over $E$ but now could be linearly dependent, is there a way to be sure they are independent?

Answer 1

If we have a basis $\{\alpha_i\}_{i=1}^n \subset K$ for the extension $K/F$, then of course we can also write every element of $K$ in the form $\displaystyle \sum_k c_k \alpha_k$ where $c_k \in E$, since $E \supset F$, meaning this is also a spanning set for $K/E$. As you noted, this set $\{\alpha_i\}_{i=1}^n$ may no longer be linearly independent when viewed as vectors over $E$. However, you can prove that some subset of $\{\alpha_i\}_{i=1}^n$ is both linearly independent over $E$ and spans $K$. The cardinality of this subset is finite and equal to the degree of $K/E$.

These considerations lay the groundwork for proving the multiplicativity formula for degrees. If $K/F$ is a finite extension, and if we have an intermediate field $E$ between $F$ and $K$, then $[K:F] = [K:E] \cdot [E:F]$.

Answer 2

If a set of $n$ elements generates a vector space $V$, then $\dim V\le n$, which is enough to conclude.

Answer 3

Yes, because every subspace of a finite dimensional vector space has finite dimension.