Noncrossed products over \(k_{\mathfrak p}(t)\) (Q2701698)

Let \(k(t)\) be a rational function field in one indeterminate \(t\) over an algebraic number field \(k\), \(p\) a prime number, \(\tilde p\) a maximal ideal of the ring of integers in \(k\) containing \(p\), \(k_{\tilde p}\) the \(\tilde p\)-adic number field, \(v_{\tilde p}\) the valuation of \(k(t)\) acting trivially on \(t\) and inducing on \(k\) its \(\tilde p\)-adic valuation, and \(k(t)_{\tilde p}\) the completion of \(k(t)\) (with respect to the topology induced by \(v_{\tilde p}\)). Also, let \(E_{\tilde p}\) be an extension of \(k_{\tilde p}\) obtained by adjoining a primitive \(l\)-th root of unity \(\varepsilon\), for some prime number \(l\neq p\), and \(E\) the extension of \(k\) generated by \(\varepsilon\). The paper under review shows that if \(\varepsilon\not\in k\), the degrees \([E:k]\) and \([E_{\tilde p}:k_{\tilde p}]\) are equal, and the groups of unity of \(l\)-primary degrees in \(E\) and in \(E_{\tilde p}\) are of one and the same order \(l^r\), then there exists a noncrossed product central division \(k(t)_{\tilde p}\)-algebra \(D_{\tilde p}\) of Schur index and exponent \(l^2r\) as well as a central division \(k(t)\)-algebra \(D\) for which there exists a \(k(t)_{\tilde p}\)-isomorphism \(D\otimes_{k(t)}k(t)_{\tilde p}\cong D_{\tilde p}\). Identifying the rational function field \(k_{\tilde p}(t)\) with the closure of \(k(t)\) in \(k(t)_{\tilde p}\), the author obtains that \(D\otimes_{k(t)}k_{\tilde p}(t)\) is a noncrossed product central division \(k_{\tilde p}(t)\)-algebra, which answers an open problem posed by \textit{D. J. Saltman} [see J. Ramanujan Math. Soc. 12, No. 1, 25-47 (1997; Zbl 0902.16021)].NEWLINENEWLINENEWLINEReviewer's remark: It follows from \textit{N. Chebotarev}'s density theorem [Math. Ann. 95, 191-228 (1925; JFM 51.0149.04)] that the main result of the paper considered applies to infinitely many prime ideals \(\tilde p_l\) of the ring of algebraic integers in \(k\). Therefore, it can be regarded as a step towards solving the non/crossed product problem for central division algebras over function fields of algebraic curves defined over \(k\).