Galois module structure
We begin with some motivating and some historical remarks.
Let N/M be a Galois extension of number fields with Galois group G. Then, according to the normal basis theorem, N is a rank one module over M[G], i.e.
with some element n ∈ N and the operation of M[G] on N being defined by
One may ask whether an analogous result also holds for the ring of integers N with M[G] being replaced by the associated order
which means that ...