Near-field (mathematics)

A fast body is an algebraic structure that serves as the coordinate area and affinity for certain projective translation levels in the synthetic geometry. It generalizes the notion of skew fields inasmuch as only one of the distributive laws is required: the left, for a legal body the Rechtsdistributivgesetz For a Fast Links Fast body. Nearly a body that meets only one of the distributive laws is also referred to as a real fast body. To distinguish between different meanings of the term the structures described here are sometimes called ( as of Zassenhaus ) Almost full body.

The projective planes of classes IVa.2 in the bilge Barlotti classification of projective planes can always be koordinatisiert by a genuine Links Fast body, as well as the unique ( up to isomorphism ) the level of the class IVa.3. The dual classes IVb.2 and IVb.3 can be koordinatisiert by real fast right body. In addition, you can arranged from Fast bodies by changing the multiplication, which is related to the Moulton -level design, models for arranged Ternärkörper construct koordinatisieren the planes of Lenz - class I.

On a finite Nearly body as coordinate space one can always construct a weakly affine space.

Every division ring is a fast body. Almost every body is a quasi body and thus a fortiori a Cartesian group and a Ternärkörper.

Definition

Left Fast body or short fast body is an algebraic structure on the set so that two two-digit addition and multiplication shortcuts are defined, for which:

Applies instead of Linksdistributivgesetzes the Rechtsdistributivgesetz, then one speaks of a legal body fast.

Equivalently one can define a left body almost as a left quasi body with associative multiplication. The same is true for the respective "right " structures.

Core of the fast body

As with a quasi body is also suited to a fast body the amount

Defined as the core of the fast body.

Properties and observations

  • A fast body is a near-ring in which a group.
  • It applies to all
  • The addition of a fast body is always commutative, in other words, is an abelian group.
  • The term " (complete ) Nearly body" of the geometry is between the terms " quasi- body" and " division ring ":
  • Almost every body is a quasi body and a quasi body is exactly then a fast body if and only if the associative law of multiplication in him.
  • Every division ring is a fast body and a fast body is exactly then a skew field if applicable in both his distributive.
  • Also between "quasi body" and " skew fields ", the terms " half-body " (in terms of geometry) and the sharper term to describe " Alternative Body ", both terms also structures that are no fast body and a fast body does not need a half-body and a fortiori no to be an alternative body.
  • A half body is exactly then a fast body when the associative law of multiplication applies in him, is he even a skew field. In other words, an algebraic structure, which is also half body and fast body is necessarily a skew field.
  • The previous two statements are identical in wording for " Alternative Body " instead of " half body ".
  • The core of a fast body is a skew field and the fast body is a module over its core. ( For geometric implications of this fact see affine translation plane! )
  • The core of a finite Nearly body is a finite field, by the theorem of Wedderburn. So every finite body is almost a finite vector space over such a finite field and therefore has elements ( prime ).
  • A fast body is then exactly half body - and thus also a skew field - if it coincides with its core.

Examples

  • Each course skew field and certainly every body provides an example of a fast body.
  • The neunelementige quasi body ( in the Examples section of order 9 ) in the article Ternärkörper described in more detail is an example of a finite almost right body half is not a body.
  • For every odd prime number can be in the finite field with elements that modify the multiplication so that a "real " full links Fast field of order arises, which is a two-dimensional vector space over its core. One possible construction is described in detail in the article Quasi body section quasi finite body Moulton levels. In order to satisfy the associative law of multiplication, one of the modified multiplication can there lay the involutory Körperautomorphismus based and so receives a Links Fast body of the type described
Hans Zassenhaus Moulton plane Wedderburn's little theorem Digital Object Identifier Bernhard Neumann Emanuel Sperner American Mathematical Society
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