Division algebras whose \(p^ i\)th powers have arbitrary index (Q1327051)

For any central simple algebra \(A\), let \(\text{ind}(A)\) denote the (Schur) index of \(A\), i.e. the degree of the division algebra which is Brauer-equivalent to \(A\), and denote by \(A^ i\) the tensor product of \(i\) copies of \(A\), for any integer \(i\). Let \(p\) be an arbitrary prime and let \(n_ k>n_{k-1}>\cdots>n_ 1>n_ 0=0\) be an arbitrary sequence of integers. The author constructs a field \(F\) and a central division \(F\)- algebra \(D\) such that \(\text{ind}(D^{{p^ \ell}})=p^{n_{k-\ell}}\) for \(\ell=0,\ldots,k\). Such an example has been first constructed by \textit{A. Schofield} and \textit{M. Van den Bergh} [Trans. Am. Math. Soc. 333, No. 2, 729-739 (1992; Zbl 0778.12004)] by different methods. The construction given here is very explicit -- the division algebra \(D\) is indeed a symbol algebra -- and the center \(F\) is a rational function field over an arbitrary base field containing sufficiently many roots of unity. The methods of proof are mostly valuation-theoretic.