On Frobenius, Mazur, and Gelfand-Mazur theorems on division algebras

Abstract

The main purpose of this paper is to introduce the reader to the theory of normed division algebras. It is written with a graduate student of mathematics in mind, who only had exposure to Advanced Calculus and to an introduction to Functional Analysis. We establish the equivalence of the Mazur and Gelfand-Mazur theorems using only elementary properties of normed algebras. Mazur's theorem [19] states that every normed division algebra over the field of real numbers is isomorphic to either the field R of real numbers, the field C of complex numbers, or the non-commutative algebra Q of quaternions. Gelfand [15] proved that every normed division algebra over the field C is isomorphic to C. He named this theorem, which is fundamental for the development of the theory of Banach Algebras, the Gelfand-Mazur theorem. An analysis of Mazur's proof (see [27], p. 18–23) shows that he proved that theorem firstly for algebras over C (using the theory of Analytic Functions) and then generalized the proof to the case of algebras over R (using the theory of Harmonic Functions.) We prove the theorems that are essential for the development in detail. These include the construction of the algebra of quaternions and the theorems of Frobenius [14], Arens [1], and Stone [26].