Cyclotomic field

Cyclotomic fields are objects of study of the mathematical part of the area of algebraic number theory. They are in some respects particularly simple generalizations of the field of rational numbers.

Properties

• Is a primitive root of unity, then the minimal polynomial of the -th cyclotomic polynomial, so is

• Two cyclotomic fields and are equal if and only if is odd and is valid.

• The adjunction of roots of unity to results with

• The extension is Galois. The Galois group is isomorphic to a primitive root of unity, this corresponds to an element by

• The Wholeness of ring is with any primitive th root of unity

• In particular, the wholeness of ring equal to the ring of integers Gaussian numbers, the wholeness of ring is equal to the ring of Eisenstein numbers. These two number fields are the only algebraic extensions of the rational numbers that are both cyclotomic and quadratic extension field.

• A prime number is accurate then branched into when a divisor of is. if and only fully disassembled when true.

• Is a prime power and a primitive root of unity, then branched into pure, and the prime ideal above is a principal ideal generated by:

Set of Kronecker -Weber

The set of Kronecker -Weber states ( by L. Kronecker and H. Weber) that every algebraic number is included with abelian Galois group in a cyclotomic field. The maximum abelian extension of that is produced by adjoining all roots of unity.