Generalized Vandermonde determinants and roots of unity of prime order

Easy proofs are given for two theorems of 0. H. Mitchell about a type of generalized Vandermonde determinant. One of these results is then used to prove that if IF(e): FI = n where F is a field of characteristic zero and e is a root of unity of prime order, then every set of n powers of e forms an F-basis for F(e).