Generalized Vandermonde determinants and roots of unity of prime order

Abstract:

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