cyclotomic field

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• Cyclotomic field — In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers. The n th cyclotomic field Q(ζn) (with n > 2) is obtained by adjoining a primitive n… …   Wikipedia

• Cyclotomic unit — In mathematics, a cyclotomic unit is a unit of an algebraic number field of the form (ζn − 1)/(ζ − 1) for ζ a root of unity, or more generally a unit that can be written as a product of these and a root of unity. The… …   Wikipedia

• Cyclotomic polynomial — In algebra, the nth cyclotomic polynomial, for any positive integer n, is the monic polynomial: where the product is over all nth primitive roots of unity ω in a field, i.e. all the complex numbers ω of order n. Contents 1 Properties …   Wikipedia

• Glossary of field theory — Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) Definition of a field A field is a commutative ring… …   Wikipedia

• Quadratic field — In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q(√d) is a bijection from the set of all square free integers d ≠ 0, 1 to the set of… …   Wikipedia

• Cubic field — In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three. Contents 1 Definition 2 Examples 3 Galois closure 4 …   Wikipedia

• Discriminant of an algebraic number field — A fundamental domain of the ring of integers of the field K obtained from Q by adjoining a root of x3 − x2 − 2x + 1. This fundamental domain sits inside K ⊗QR. The discriminant of K is 49 = 72.… …   Wikipedia

• Algebraic number field — In mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector… …   Wikipedia

• CM-field — In mathematics, a CM field is a particular type of number field K , so named for a close connection to the theory of complex multiplication. Another name used is J field . Specifically, K is a totally imaginary quadratic extension of a totally… …   Wikipedia

• Algebraically closed field — In mathematics, a field F is said to be algebraically closed if every polynomial in one variable of degree at least 1, with coefficients in F , has a root in F . ExamplesAs an example, the field of real numbers is not algebraically closed,… …   Wikipedia

• Class field theory — In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields. Most of the central results in this area were proved in the period between 1900 and 1950. The theory takes its name… …   Wikipedia