Wedderburn's little theorem

The set of Wedderburn ( by Joseph Wedderburn ) belongs to the mathematical field of algebra. He says that every finite division ring is a field, that is, if a skew field contains only a finite number of elements, it follows automatically the commutativity of multiplication. In other words, every skew field which is not a body ( where the multiplication is not commutative ie ) must contain infinitely many elements.

In addition to Wedderburn ( the more evidence was, first 1905) have also supplied other mathematicians different proofs of the theorem, for example, Leonard Dickson, Emil Artin, Ernst Witt (the proof includes a page ), Hans Zassenhaus, Israel Herstein.

There are other famous phrases that are sometimes called simple set of Wedderburn, as his sentence for the classification of semisimple algebras, generalizing the set of Artin - Wedderburn. In English Wedderburns set is called over finite skew field therefore Wedderburn 's Little Theorem.

Application

This theorem has an important application in synthetic geometry: For finite affine or projective planes follows from the theorem of Desargues, the set of Pappus. It can be used as affine or projective plane consider each desarguesche level over a skew field K, the set of Pappus if and only if K is commutative. Here the set of Wedderburn is used. For these purely geometrical facts known to date no geometric proof.

The inverse proposition: Each level is pappossche desarguessch is as a set of Hessenberg called (after Gerhard Hessenberg ) and applies to each and every affine projective plane.