Artin–Zorn theorem

In mathematics, the Artin–Zorn theorem is an important idea named after two mathematicians, Emil Artin and Max Zorn. This theorem tells us that any small, special kind of number system, called a finite alternative division ring, must actually be a more common and well-studied number system known as a finite field.

The theorem was first shared in 1930 by Zorn, but he gave credit to Artin for the idea. It builds on another famous result called Wedderburn's little theorem, which says that certain number systems with special rules are also fields. One interesting result from the Artin–Zorn theorem is that a special kind of geometric space, called a finite Moufang plane, is actually just a basic projective plane created using a finite field. This theorem helps mathematicians understand the structure and limits of number systems and their connections to geometry.