Mal'cev-Neumann rings and noncrossed product division algebras. (Q2892993)

A finite dimensional division algebra is a crossed product if it contains a maximal subfield over the center. In 1972 Amitsur showed that for degree divisible by an odd square or by \(8\), the generic division algebra is a noncrossed product. As part of the theory of central simple algebras over special fields, there is interest in constructing more explicit noncrossed products. Several examples were given, most notably by Brussel and Hanke. In this paper, Coyette shows that Malcev-Neumann algebras associated to generalized (Abelian) crossed products are another interesting source of noncrossed products.NEWLINENEWLINE Let \(A\) be a central simple algebra over a field \(F\), and assume \(A\) contains a (nonmaximal) field \(F'\), Galois over \(F\), with an Abelian Galois group \(G=\text{Gal}(F'/F)\). Let \(A'=C_A(F')\) denote the centralizer of \(F'\) in \(A\). Fix, for each \(\sigma\in G\), an element \(u_\sigma\in A\) such that \(u_\sigma^{-1}xu_\sigma=\sigma(x)\) for \(x\in F'\). Let \(\Gamma\) be a totally ordered Abelian group with a surjection \(\varepsilon\colon\Gamma\to G\). The Malcev-Neumann algebra \(\mathcal D=A'((\Gamma))\) is the ring of formal series \(\sum_{\gamma\in\Gamma}a_\gamma r_\gamma\) in the variables \(a_\gamma\) with coefficients \(r_\gamma\in A'\) and with well ordered support, subject to the relations \(r(a_\gamma u_{\varepsilon(\gamma)}^{-1})=(a_\gamma u_{\varepsilon(\gamma)}^{-1})r\) for \(r\in A'\) and \(a_\gamma a_\delta=a_{\gamma+\delta}u^{-1}_{\varepsilon(\gamma+\delta)}u_{\varepsilon(\gamma)}u_{\varepsilon(\delta)}\). For \(\Gamma=\mathbb Z\) one obtains a twisted Laurent series ring in one variable over \(A'\).NEWLINENEWLINE Assume \(A'\) is a division algebra. Then \(\mathcal D\) is a valued division algebra with value group \(\Gamma\) and residue ring \(A'\), whose center is \(F((\Gamma_0))\) for \(\Gamma_0=\ker(\varepsilon)\), and with degree equal to the degree of \(A\). The key result is that Galois maximal subfields of \(\mathcal D\) give rise to Galois maximal subfields of \(A\) which contain \(F'\), so if \(A\) has no maximal Galois subfields \textit{containing} \(F'\), then \(\mathcal D\) is a noncrossed product.NEWLINENEWLINE Local invariants are then used to complete the construction, producing noncrossed products of degree \(p^2\) whose center is \(\mathbb Q((t))\) for any odd prime \(p\), and a noncrossed product of degree \(8\) whose center is \(\mathbb Q((t_1,t_2))\).