Wedderburn’s factorization theorem application to reduced $K$-theory

Abstract

Abstract. This article provides a short and elementary proof of the key theorem of reduced K-theory, namely Platonov's Congruence theorem. Our proof is based on Wedderburn's factorization theorem.Let D be a division algebra with center F . If a ∈ D is algebraic over F of degree m, then by Wedderburn's factorization theorem one can find m conjugates of a such that the sum and the product of them are in F . This observation has been used in many different circumstances to give a short proof of known theorems of central simple algebras. (See [8] for a list of these theorems.) Here we will use this fact to prove Platonov's congruence theorem.The non-triviality of the reduced Whitehead group SK 1 (D) was first shown by V. P. Platonov who developed a so-called reduced K-theory to compute SK 1 (D) for certain division algebras. The key step in his theory is the "congruence theorem" which is used to connect SK 1 (D), where D is a residue division algebra of D to SK 1 (D). This in effect enables one to compute the group SK 1 (D) for certain division algebras. (See [5] and [6].)Before we describe the congruence theorem, we employ Wedderburn's factorization theorem to obtain a result regarding normal subgroups of division algebras.