Every Finite Division Ring is a Field
Friday 22nd January, 2010 16:00-17:00 Mathematics Building, room 515
It is easy to think of a non-commutative division ring: the quaternions. However, as the title suggests, we can prove that any non-commutative division ring must necessarily be infinite. Many mathematicians (Artin, Bourbaki, Wedderburn and others) have given proofs of this remarkable theorem. We present a particularly elegant offering given by Ernst Witt in 1931. Only elementary ideas are needed: thus, the only prerequisites are linear algebra, ultra basic group theory and familiarity with complex roots of unity. Briefly perusing the class formula may be advantageous.