## A fact about finitely generated field extensions

Let $F$ be a field and let $E$ be an extension of $F$. Suppose $E$ is finitely generated as an $F$-vector space; say by $B = \{b_i\}_{i=1}^n$. ($B$ need not be a basis.) Prove that $E$ is a finite extension of $F$. What can we say about the degree of $E$ over $F$?

Recall that every finite generating set of a vector space contains a basis. Thus there is a subset $B^\prime \subseteq B$ which is a basis for $E$ over $F$. In particular, $E$ has finite dimension as an $F$-vector space. Moreover, the dimension of $E$ (i.e. its degree over $F$) is at most $n$.