Order (ring theory)

In algebraic number theory is an order of the number field K is a subring of K (via multiplication) as endomorphism on certain subsets of K, the grids operated on at the same time is the order itself, a special grid. The terms order and lattice play a role in the investigation of Teilbarkeitsfragen in number fields and in the generalization of the fundamental theorem of arithmetic to number fields. These ideas and conceptions are due to Richard Dedekind. The more specific definitions in the first part of the article are based on Leutbecher (1996). Then, a generalization of the notion of Procedure by Silverman (1986 ) is described. To distinguish them from the more general and more specific terms, the different ligands terms are also called Dedekind lattice and Dedekind order.

Definitions

  • A number field K is here an extension field of the field of rational numbers, which has a finite dimension n over the rational numbers. This dimension is called the degree of the field extension.
  • As a lattice in a number field K is called every finitely generated subgroup of M (K, ), which contains a base of K.
  • Two grids M and N are (broadly defined ) are equivalent if there is a number that goes with the λ · M = N, in the strict sense equivalent if such a λ exists even in.
  • The order of a lattice M is. Equivalent to this is: Each grid G, which is also a subring of K, is an order (and at least of himself as a grid, but in addition also of all equivalent lattices ).

Properties

  • Equivalents grid have the same order.
  • Each order is itself a lattice.
  • Each order is a subring of K.
  • Each element of an order is an algebraic integer, and thus also an algebraic number.
  • Is algebraic and quite an order, then is also an order.
  • There exists a K in the sense of inclusion maximal order, the principal order or maximal order of K.
  • The principal order includes exactly all algebraic integers in K, that is, the terms wholeness ring and principal order refer to the same subset of K.

Related to geometric lattices

The wording grid indicates a connection with lattices in Euclidean spaces, the fact is: The number field K is an n-dimensional vector space over. This vector space can be embedded into an n - dimensional real vector space. In this vector space are the Dedekind lattice specific geometric grid. Dedekind lattice are never "flat" (that is, in a real subspace included), as they must contain a basis of K and thus always in a real vector space basis.

The vivid imagination of a lattice in n-dimensional space can be useful for understanding. For example, for an integer k> 1, the Dedekind lattice k · M a grid as the Dedekind lattice M is " coarse-mesh ". The lattice M and k · M can be mapped to each other by central dilations.

In evidence, which refer to the embedding described with respect, caution is advised. If, for example, in a number field K that contains the algebraic number, multiplies it as a vector scalar with the real number, then the result is not "2". In order to distinguish the different multiplications, these must be formally correct as embedding tensor

Introduce (see the next section).

Generalization

More generally, a finite, not necessarily commutative algebra, it is called a subring an order in when

  • A finitely generated module and
  • The canonical homomorphism

This notion generalizes the concept of order in a number field defined above. Examples of orders in quaternion algebras over endomorphism rings are super singular elliptic curves.

Integer factorization Ring of integers Tensor product Algebra over a field
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