Ring of integers

In the mathematical subfield of algebraic number theory is the ring of an algebraic number field is the analogue of the ring of integers in the case of the field of rational numbers. The elements of a whole ring are referred to as algebraic integers, the set of all algebraic integers is the is the ring of all algebraic numbers in the body.

Definition

It is an algebraic number field, that is, a finite extension of the field of rational numbers. Then the totality of ring is defined as the entire completion of in, ie the subset of those satisfying an equation of the form

Comply with. Note that the coefficient of ( the leading coefficient of the polynomial ) must be equal to 1. We call such polynomials to be normalized. Without these restrictions you would get the whole body.

An equivalent definition is: The whole ring of K is the maximal order in the sense of inclusion, the principal order on K.

Properties

• Is a Dedekind ring.

Examples

• If so is the ring of Eisenstein numbers

• If so is the ring around the Gaussian numbers.

• General looks to the totality of the ring ( which was quite and square-free ) an integral basis as follows:

• Describes a primitive root of unity, the wholeness of the ring -th cyclotomic field is the same.

• If the skew field of rational quaternions (from Latin quaternio " Tetrad " )