Let be a field and let be an extension of . Suppose is finitely generated as an -vector space; say by . ( need not be a basis.) Prove that is a finite extension of . What can we say about the degree of over ?
Recall that every finite generating set of a vector space contains a basis. Thus there is a subset which is a basis for over . In particular, has finite dimension as an -vector space. Moreover, the dimension of (i.e. its degree over ) is at most .