Algebraic number field

An algebraic number field or a number of short body referred to in mathematics a finite extension of the field of rational numbers. The study of algebraic number fields is a central topic of algebraic number theory, a branch of number theory.

Play an important role in the totality of rings of algebraic number fields, which are analogues of the ring of integers in the body.

Definition and simple properties

An algebraic number field is defined as a finite field extension of the field of rational numbers. This means that a vector space having a finite dimension. This dimension is called degree of the number field.

As a finite number field extensions are always algebraic extensions, that is, each element of a number field is zero of a polynomial with rational coefficients and is therefore an algebraic number. Conversely, however, not every algebraic extension of a number field: For example, the bodies of all algebraic numbers is indeed an algebraic, but not a finite extension of, so no algebraic number field.

By the theorem of the primitive element number field are simple extensions of the body, so can be in the form to be represented as adjunction of an algebraic number.

Wholeness

An element of a number field is completely known, if it is zero of a normalized polynomial ( leading coefficient 1) with coefficients from. That is, satisfies an equation of the form

Integers. Such numbers are called ganzalgebraische numbers.

The integers form a subring of, the totality of ring is called, and usually with, or is also referred to.

Examples

  • As a trivial example itself is a number field ( of degree 1 ). As expected valid, i.e. that all rational numbers are the "normal" integers.
  • The field of complex numbers with rational real and imaginary parts is a number field of degree 2 The associated wholeness ring is the ring of the ( whole ) Gaussian numbers.
  • General form the quadratic number fields with square -free exactly the number field of degree 2 for the wholeness rings results
  • The cyclotomic field with a primitive -th root of unity is a number field of degree with the Euler's φ function. The whole ring.

Bases

Since a number field of degree one -dimensional vector space, each based on exactly elements. If such a basis, then each element can be written in the form

With uniquely determined coefficients, however, depend on the choice of the base. Applies only then has the specific base, the degree of equal to the degree of the minimal polynomial of the algebraic number.

A basis of is called an integral basis if each entire element can be written in the form with. For example, a base of, but no integral basis, because not all elements of the whole ring can be used as integer linear combinations of 1 and write. In contrast, an integral basis of.

Another basis dependent representation of elements of a number field is the matrix representation. Be elected to fixed, then is given by the multiplication by a linear map. This endomorphism can be represented with respect to a fixed base by a square matrix. The determinant and the trace of the figure (ie, the representing matrix ), which are independent of the choice of Basic, Standard or track of are mentioned and are important tools for invoices and proofs in algebraic number fields.

Generalization and classification

The algebraic number field, together with the function fields of characteristic class of global bodies, which are among those about the body of the p- adic numbers, together with the local twills, which represent the main research objects of algebraic number theory.

References

  • Ideal class group
  • Dirichlet's unit theorem
  • Kummer extension
  • Quadratic reciprocity, reciprocity Artinsches
  • Class field theory
  • Brauer group
  • Iwasawa theory
  • Dedekind zeta function
Adjunction (field theory) Square-free integer Minimal polynomial (linear algebra) Transformation matrix Characteristic (algebra) Dirichlet's unit theorem Kummer theory
48040
de