Wedderburn theorem

The set of Wedderburn (by Joseph Wedderburn ) belongs to the mathematical branch of algebra . It says that every finite oblique body is a field , that is, if a oblique body only contains finitely many elements, the commutativity of the multiplication follows automatically . In other words: every oblique body that is not a field (in which the multiplication is not commutative) must contain an infinite number of elements.

In addition to Wedderburn (who gave several proofs, first in 1905), other mathematicians have also provided different proofs for the theorem, for example Leonard Dickson , Emil Artin , Ernst Witt (the proof covers one page), Hans Zassenhaus , Israel Herstein .

There are other well-known theorems, sometimes called simply Wedderburn's Theorem, such as his Theorem on the Classification of Semi-Simple Algebras, generalized in Artin-Wedderburn's Theorem . In English, Wedderburn's theorem about finite skew bodies is therefore also called Wedderburn's Little Theorem .

application

This set has an important application in synthetic geometry : For finite affine or projective planes follows from the set of Desargues the set of Pappos . Every Desargue's plane can be viewed as an affine or projective plane over an inclined body K , where Pappos's theorem applies if and only if K is commutative. This is where Wedderburn's theorem comes into play. To this day, no geometric proof is known for this purely geometric situation.

The opposite statement: every Pappos plane is desarguessic is called Hessenberg's theorem (after Gerhard Hessenberg ) and applies to every affine and every projective plane.

literature

Kurt Meyberg: Algebra. Part 2. Hanser, Munich 1976, ISBN 3-446-12172-2 , p. 63.
Martin Aigner , Günter M. Ziegler : Proofs from the book. Springer, Berlin et al. 1998, ISBN 3-540-63698-6 .

Individual evidence

↑ The first one was faulty. The history of the evidence has been examined by Karen Parshall .
^ Ernst Witt, On the commutativity of finite skew bodies , treatises from the Mathematical Seminar of the University of Hamburg, Volume 8, 1931, p. 413, doi : 10.1007 / BF02941019
↑ For example Bartel L. van der Waerden , Algebra , Volume 2, Springer, Heidelberger Taschenbücher, p. 73
^ ^A ^b ^c Heinz Lüneburg : The Euclidean plane and its relatives . Birkhäuser, Basel / Boston / Berlin 1999, ISBN 3-7643-5685-5 , III: Papossche Ebene ( digitized reading sample from google-books [accessed on July 30, 2013] Detailed discussion and proof of the Hessenberg theorem, explanations, such as the Pappos' theorem determines the algebraic structure of the coordinate field).

[1] The first one was faulty. The history of the evidence has been examined by Karen Parshall .

[2] Ernst Witt, On the commutativity of finite skew bodies , treatises from the Mathematical Seminar of the University of Hamburg, Volume 8, 1931, p. 413, doi : 10.1007 / BF02941019

[3] For example Bartel L. van der Waerden , Algebra , Volume 2, Springer, Heidelberger Taschenbücher, p. 73

[LBHessenberg-4] A ^b ^c Heinz Lüneburg : The Euclidean plane and its relatives . Birkhäuser, Basel / Boston / Berlin 1999, ISBN 3-7643-5685-5 , III: Papossche Ebene ( digitized reading sample from google-books [accessed on July 30, 2013] Detailed discussion and proof of the Hessenberg theorem, explanations, such as the Pappos' theorem determines the algebraic structure of the coordinate field).