Titlebook: Classical Theory of Algebraic Numbers; Paulo Ribenboim Textbook 2001Latest edition Springer Science+Business Media New York 2001 algebra.a

外来

Extort

污点

Commutative FieldsFor the convenience of the reader we recall some definitions and facts about commutative fields.

Anguish

Residue ClassesIn this chapter, we study residue classes modulo a natural number. This leads to the consideration of groups. Therefore it is convenient to recall that if . is a finite group, the number of elements of . is called the . of ., denoted by ..

配置

goodwill

Algebraic IntegersThe arithmetic of the field of rational numbers is mainly the study of divisibility properties with respect to the ring of integers.

贸易

Integral Basis, DiscriminantWe have seen in the numerical examples of the preceding chapter that the ring of algebraic integers of a quadratic number field, and also of the cyclotomic field ℚ(ζ) (where ζ is a primitive .th root of unity), are free finitely generated Abelian groups.

Obstruction

The Decomposition of IdealsWe have shown that the ring . of algebraic integers of an algebraic number field is Noetherian and integrally closed. However, it is not true in general that . is a principal ideal domain.

ADJ

The Norm and Classes of IdealsWe know already that the ring . of integers of an algebraic number field . need not be a principal ideal domain. In this chapter, we associate with every field . a numerical invariant ., which measures the extent to which . deviates from being a principal ideal domain. . will be equal to 1 if and only if . is a principal ideal domain.

OGLE

Estimates for the DiscriminantIn this chapter we study the discriminant. A method of “Geometry of Numbers” is used to provide sharper estimates for the discriminant.