Inverse Galois problem over the skew field of rational fractions with central indeterminate (Q2301891)

In this paper, the authors introduce the problem inverse of Galois over a division algebra \(H\) of finite dimension over its center \(k\) (\(PIG_H\)); that is for a finite group \(G\), is there a finite extension \(L/K\) of division algebras such that \(G\) is the set of \(H\)-isomorphisms of \(L\). The authors give an example where the implication \(PIG_H \Longrightarrow PIG_k\) is false, and so they focus on its converse, namely under which conditions the implication \(PIG_k \Longrightarrow PIG_H\) is true. They used the reduced norm of a finite dimension division algebra over its center \(k\), to characterize the linearly disjoint field extensions. They also introduce a problem inverse of Galois with constraints (\(PIGFC_k\)), and they prove that if the constraint \(F\) is defined by the reduced norm of the extension \(H/k\), then \(PIGFC_k \Longleftrightarrow PIG_H\). In particular, if \(k(t)\) is the field of rational fractions over \(k\) and \(H(t)\) is the skew field of rational fractions with central indeterminate, then \(PIGFC_{k(t)} \Longleftrightarrow PIG_{H(t)}\). The authors achieve their result by showing that if the field \(k\) contains an ample field and \(H\) is a finite dimensional division algebra over its center \(k\), then the problem \(PIG_{H(t)}\) has a positive answer.