Wedderburn's factorization theorem. Application to reduced \(K\)-theory (Q2758964)

A valuation \(v\) on a field \(F\) is said to be Henselian, if it is uniquely extendable to a valuation on each algebraic extension of \(F\). When this occurs, \(v\) extends uniquely to a valuation (also denoted by \(v\)) of each central division \(F\)-algebra \(D\) of finite dimension. Assuming that \(v\) is Henselian, denote by \(V_D\), \(V_F\) the valuation rings of \(v\) on \(D\) and \(F\), respectively, by \(M_D\), \(M_F\) their maximal ideals, and by \(\overline D\), \(\overline F\) their residue division algebra and residue field, respectively. We say that \(D\) is a tame division \(F\)-algebra, if the Schur index \(\text{ind}(D)\) is not divisible by \(\text{char}(\overline F)\). Platonov's congruence theorem asserts that if \(D\) is tame, \(D^1\) is the kernel of the reduced norm, and \(D'\) is the commutator subgroup of \(D^*\), then \((1+M_D)\cap D^1\subseteq D'\). It has been proved by Platonov in the special case, where \((F,v)\) is complete and discrete valued, and by Ershov in full generality [see \textit{V. P. Platonov}, Izv. Akad. Nauk SSSR, Ser. Mat. 40, 227-261 (1976; Zbl 0338.16005); \textit{Yu. L. Ershov}, Mat. Sb., Nov. Ser. 117(159), 60-68 (1982; Zbl 0508.16017); \textit{A. A. Suslin}, Adv. Sov. Math. 4, 75-99 (1991; Zbl 0758.11047)]. Used for connecting the reduced Whitehead groups \(SK_1(D)\) and \(SK_1(\overline D)\), the theorem has been the crucial step in the development of the reduced \(K\)-theory, which enables one to compute the reduced Whitehead group for certain division algebras, and shows that the nontriviality of these groups is a usual phenomenon [see \textit{V. P. Platonov, V. I. Yanchevskij}, Algebra IX, Finite dimensional division algebras, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravl. 77, 144-262 (1991; Zbl 0804.16019); Engl. transl. in Encycl. Math. Sci. 77, Springer, Berlin (1996)]. The so far existing proofs of this result are involved and lengthy.NEWLINENEWLINENEWLINEThe paper under review provides a short and elementary proof of the congruence theorem, based on Wedderburn's factorization theorem, i.e. the statement that if \(D\) is a central division algebra over an arbitrary field \(F\), and \(a\) is an element of \(D\) algebraic over \(F\) of degree \(m\), then the minimal polynomial \(f(X)\) of \(a\) over \(F\) is equal to the product \(\prod_{i=1}^m(X-d_i\cdot a\cdot d_i^{-1})\), for some \(d_i\in D^*\). The author's main lemma is of independent interest. It states that if \(D\) is of index \(n\) over \(F\), and \(N\) is a normal subgroup of \(D^*\), then \(N^n\subseteq(F^*\cap N)\cdot[D^*,N]\).