Let \(F\) be a field with a valuation \(v\) and let \(D\) be a central division \(F\)-algebra of finite dimension \([D:F]\). The conditions under which \(v\) extends to a valuation of \(D\) have been well-understood 15-20 years ago by Wadsworth, Ershov and the first author.
The paper under review deals with the problem of whether or not \(v\) is extendable to a valuation \(v_i\) on the underlying division \(F\)-algebra of the \(i\)-th tensor power of \(D\) over \(F\), for \(i=1,\dots,[D:F]-1\). This problem standardly reduces to the special case where \(D\) is of exponent \(p^m\), for some prime number \(p\) and some positive integer \(m\), and for \(i=p^j\); \(j=1,\dots,m -1\).
The authors prove that the (Schur) index of \(D\) is divisible by \(p^{m+d}\), where \(d\) is the number of “no-yes” subpatterns in the sequence of answers for \(D/(F,v)\). Also, they show constructively that if \(S=(s_m,\dots,s_1)\) is a sequence of “yes’s” and “no’s”, and \(\delta\) is the number of “no-yes” subpatterns in \(S\), then one can find a valued field \((F(S),v(S))\) and a central division \(F(S)\)-algebra \(D(S)\) of index \(p^{m+\delta}\) and with sequence of answers \(S\).