Symbol length and stability index
Let \(F\) be a field of characteristic not \(2\). Let \(n\in{\mathbb N}\) and denote by \(k_n(F)\) the Milnor \(K\)-group \(K_n(F)\) modulo \(2\). \(k_n(F)\) is additively generated by ``symbols'' \(\{ a_1,\ldots,a_n\}\), \(a_i\in F^\times\), and by theorems by Voevodsky resp. Orlov-Vishik-Voevodsky it is known that \(k_n(F)\) is isomorphic via canonical homomorphisms to \(I^nF/I^{n+1}F\) (where \(I^nF\) denotes the \(n\)-th power of the fundamental ideal \(IF\) in the Witt ring \(WF\)) and to \(H^n(F,{\mathbb Z}/2{\mathbb Z})\), the \(n\)-th Galois cohomology group of \(F\) with coefficients in \({\mathbb Z}/2{\mathbb Z}\) (these statements were formerly known as the Milnor Conjectures). The \(n\)-symbol length \(\lambda_n(F)\) is defined to be the smallest nonnegative integer \(m\) such that each element in \(k_n(F)\) can be written as a sum of \(m\) symbols, provided such an integer exists, and \(\lambda_n(F)=\infty\) otherwise. If \(F\) is real and \(T\) is a preordering on \(F\) with \([F^\times:T^\times]=2^{n+1}\), and \(X_T\) is the space of those orderings on \(F\) that contain \(T\), then it is known that \(|X_T|\leq 2^n\), and \(T\) is called a fan of degree \(n\) if equality holds. The (reduced) stability index \(\text{st}(F)\) is defined to be the supremum of the degrees of all the fans of \(F\).NEWLINENEWLINEThe main purpose of the present paper is to relate \(\text{st}(F)\) to \(\lambda_n(F)\) for a real field \(F\), and more generally in the context of an abstract Witt ring with suitable adaptations of the definitions of the symbol length and the stability index. The results for fields read as follows. If \(\lambda_i(F)<\infty\) for some \(i\geq 2\) then \(\text{st}(F)<\infty\), in particular \(\text{st}(F)\leq 2\lambda_2(F)-1\). If \(F\) is Pythagorean, then one also has that \(s=\text{st}(F)<\infty\) implies \(\lambda_2(F)<\infty\). More precisely, one has \(\lambda_2(F)=s\) if \(1\leq s\leq 2\), and \([s/2]+1\leq \lambda_2(F)\leq 2^{s-1}(2^{s-2}-1)\) if \(s\geq 3\). Examples show that for \(s\geq 3\), the lower bound is sharp, but it is apparently not clear how good the upper bound really is.
scientific article; zbMATH DE number 6078362
scientific article