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A Comprehensive Course in Number Theory by Alan Baker

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10

Number fields

10.1 Introduction

The subject of number fields was originated by Kummer, Dedekind, Weber and others during the nineteenth century. It is closely related to the theory of Diophantine equations – in fact it was motivated to a large extent by attempts to solve Fermat’s last theorem – and it now impinges on most branches of number theory. We have already given a short discussion of the topic, mainly with respect to quadratic fields, in Chapter 7; here and in the next three chapters we shall develop the subject in more detail.

As prerequisites we shall assume only the elementary properties of rings, fields and vector spaces. Some knowledge of Galois theory is useful but, as we shall present the material, not essential. However, to ...

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