Noncrossed product bounds over Henselian fields. (Q399370)

A central simple algebra is a crossed product if it has a maximal subfield which is Galois over the center. After Amitsur proved the existence of noncrossed products, and Brussel demonstrated their existence over fields such as \(\mathbb{Q}(\!(t)\!)\), the next logical step is to pinpoint explicit noncrossed products in the Brauer group we are able to compute.NEWLINENEWLINE Let \(F\) be a Henselian field, with an arbitrary value group \(\Gamma\), and assume the residue field is a global field \(K\). The (generalized) Witt decomposition describes the subgroup of inertially split classes (those split by an unramified extension), as the direct sum \(\text{SBr}(F)=\text{Br}(K)\oplus\Hom_c(G_K,\Delta/\Gamma)\), where \(G_K\) is the absolute Galois group of \(K\), and \(\Delta\) is the divisible hull of \(\Gamma\). The index of the class corresponding to \(\alpha+\chi\) is always divisible by \(|\chi|\). Following \textit{T. Hanke} and \textit{J. Sonn} [Math. Ann. 350, No. 2, 313-337 (2011; Zbl 1242.16021)], the authors study the fibers \(\text{Br}(K)+\chi\), and prove:NEWLINENEWLINE Theorem 1.1: For each \(\chi\) there is a supernatural number \(b(\chi)=\prod p^{b_p(\chi)}\) (where \(b_p(\chi)\in\{0,1,\ldots,\infty\}\)), such that if \(m\) divides \(b(\chi)\) then all division algebras of index \(m|\chi|\) in \(\text{Br}(K)+\chi\) are crossed product, but the fiber contains infinitely many noncrossed products of index \(m|\chi|\) otherwise.NEWLINENEWLINE Theorem 1.2: If \(p\neq\text{char}(K)\) and the \(p\)-Sylow of \(\text{Im}(\chi)\) is non-cyclic, then \(b_p(\chi)\) is finite; in fact \(b_p(\chi)\) is zero if the kernel of \(\chi\) does not fix \(p\)-roots of unity. Thus, the fiber \(\text{Br}(K)+\chi\) contains noncrossed products whenever the maximal prime-to-\(\text{char}(K)\) subgroup of \(\text{Im}(\chi)\) is non-cyclic.NEWLINENEWLINE The proofs start from the observation that the division algebra underlying \(\alpha+\chi\) is a crossed product if and only if the division algebra underlying \(\alpha^M\) (the image of \(\alpha\) in \(\text{Br}(M)\) where \(M=\overline K^{\text{Im}(\chi)}\)) contains a maximal subfield \(L\) which is Galois over \(K\). Such subfields are characterized by local data, restricted by the action of \(\text{Gal}(M/K)\) on \(\text{Gal}(L/M)\). Analysis of this action leads to a ``formula'' for the bounds \(b_p(\chi)\) (Corollary 2.17).