## Abstract

In this chapter we present first the progress in the study of the structure of number fields, the central subject being the existence of normal and normal integral bases, and then consider some additive questions, mainly on sums of squares. The next section concentrates on the simplification of the class-field theory by Hasse and Chevalley, and the following sections concern i.a. the class-number and class-group of quadratic fields, the question of the existence of Euclidean algorithm in fields, the distribution of algebraic integers on the complex plane and infinite extensions of number fields.

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