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.

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