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Complex Multiplication by Reinhard Schertz

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8

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 ...

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