chapter  4
32 Pages

Topological Division Rings

WithL. S. Pontryagin

Along with topological groups an important role is played in mathematics by topological rings and fields, that is, rings and fields in which the algebraic operations are continuous. A continuous division ring is defined by a small number of entirely natural axioms; nevertheless the concept possesses an extraordinary concreteness. Every connected locally compact topological division ring is isomorphic either with the field of real numbers, the field of complex numbers, or the division ring of quaternions. The most definitive result relates to the connected case. The relevant theorem, which is due to the author, asserts that every locally compact connected division ring is isomorphic with either the field of real numbers, the field of complex numbers or the division ring of quaternions.