On a Theorem of Lenstra and Schoof

We give a detailed proof of Theorem 1.15 from a well-known paper "Primitive normal bases for finite fields" by H.W. Lenstra Jr. and R.J. Schoof. We are not aware of any other proofs. Let $L/K$ be a finite-dimensional Galois field extension and $B$ the set of all normal bases of this extension. Theorem 1.15 describes the group of all $\gamma$ in the multiplicative group of $L$ such that $\gamma B = B$.