Order (algebraic number theory)

In algebraic number theory , an order of the number field is a subring of , which operates (via multiplication) as an endomorphism ring on certain subgroups of , the lattice , at the same time the order itself is a special lattice. The terms order and lattice play a role in the investigation of questions of divisibility in number fields and in the generalization of the fundamental theorem of arithmetic to number fields. These ideas and concepts go back to Richard Dedekind . The more specific definitions in the first part of the article are based on Leutbecher (1996). Then a generalization of the term order according to Silverman (1986) is described. To distinguish between more general and deviating terms, the more specific terms are also referred to as Dedekind lattice and Dedekind order . ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$

Definitions

• A number field is here an expansion field of the field of rational numbers, which has a finite dimension above the rational numbers . This dimension is called the degree of body expansion. ${\displaystyle K}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle n}$
• Every finitely generated subgroup of that contains a - basis of is called a lattice in the number field . Lattices in the free subgroups of with rank are equivalent . ${\displaystyle K}$ ${\displaystyle M}$ ${\displaystyle (K,+)}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle [K:\mathbb {Q} ]}$
• Two lattices and are called equivalent (in the broader sense) if there is a number with which applies, equivalent in the narrower sense if such a number even exists in. ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle \lambda \in K^{\times }}$ ${\displaystyle \lambda \cdot M=N}$ ${\displaystyle \lambda }$ ${\displaystyle \mathbb {Q} }$
• The order of a lattice is . Equivalent to this is: Every lattice G , which is at the same time a sub-ring of K , is an order (at least of itself as a lattice, but also of all equivalent lattices). ${\displaystyle {\mathcal {O}}}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {O}}={\mathcal {O}}(M)=\{\omega \in K:\omega \cdot M\subseteq M\}}$

properties

• Equivalent grids have the same order.
• Every order is itself a grid.
• Every order is a subring of . ${\displaystyle K}$
• Each element of an order is an algebraic integer .
• If algebraically is whole and an order, then there is also an order. ${\displaystyle \alpha \in K}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {O}}[\alpha ]}$
• It exists via a maximum order in the sense of inclusion , the main order or maximum order of . ${\displaystyle K}$ ${\displaystyle {\mathcal {O}}_{K}}$ ${\displaystyle K}$
• The main order includes exactly all algebraic integers in , i.e. H. the terms wholeness ring and main order denote the same subset of . ${\displaystyle K}$ ${\displaystyle K}$

Connection with geometric grids

The choice of words lattice indicates a connection with lattices in Euclidean spaces that actually exists: The number field is a -dimensional vector space over . This vector space can be embedded in a -dimensional real vector space. In this vector space the Dedekind lattices are special geometric lattices. Dedekind lattices are never "flat" (that is, contained in a real subspace), since they must always contain a- basis of and thus a- basis in real vector space . ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle K}$ ${\displaystyle \mathbb {R} }$

The descriptive idea of ​​a grid in -dimensional space can be useful for understanding. For example, for an integer, the Dedekind grid is a grid that is “wider mesh” than the Dedekind grid . The grids and can be mapped onto one another through centric stretching. ${\displaystyle n}$ ${\displaystyle k>1}$ ${\displaystyle k\cdot M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle k\cdot M}$

Caution should be exercised with evidence referring to the embedding described. If, for example, in a number field that contains the algebraic number , this is multiplied as a scalar vector with the real number , then the result is not . In order to distinguish the different multiplications, one has to formally correct this embedding as a tensor product ${\displaystyle K}$ ${\displaystyle {\sqrt {2}}}$ ${\displaystyle {\sqrt {2}}}$ ${\displaystyle 2}$

${\displaystyle {\mathcal {O}}\to K\otimes \mathbb {R} }$

introduce (see next section ).

generalization

If, more generally, is a finite-dimensional , not necessarily commutative - algebra , then a subring is called an order in if ${\displaystyle A}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle {\mathcal {O}}\subset A}$ ${\displaystyle A}$

• ${\displaystyle {\mathcal {O}}}$a finitely generated - module and ${\displaystyle \mathbb {Z} }$
• the canonical homomorphism

${\displaystyle {\mathcal {O}}\otimes \mathbb {Q} \to A}$

is an isomorphism.

This term generalizes the concept of order in a number field defined above. Examples of orders in quaternion about are endomorphism rings super singular elliptic curves . ${\displaystyle \mathbb {Q} }$

literature

• Armin Leutbecher: Number Theory. An introduction to algebra. Springer, Berlin et al. 1996, ISBN 3-540-58791-8 .
• Joseph H. Silverman: The Arithmetic of Elliptic Curves (= Graduate Texts in Mathematics. Vol. 106). Springer, New York NY 1986, ISBN 3-540-96203-4 , orders, especially in quaternion algebras: III.§9; supersingular elliptic curves: V. §3.

swell

1. ↑ PG Lejeune Dirichlet : Lectures on number theory. Edited and with additions by R. Dedekind . 4th, revised and enlarged edition. Vieweg, Braunschweig 1894.