Galois theory of semilinear transformations (Q2712786)

One can realize general linear groups \(\text{GL}(m,q)\) (with \(m\) a positive integer and \(q\) a power of a prime \(p\)) as Galois groups of vectorial (\(=q\)-additive) polynomials over rational function fields of characteristic \(p\) [see \textit{S. S. Abhyankar}, Isr. J. Math. 88, 1-23 (1994; Zbl 0828.14014), Proc. Am. Math. Soc. 124, 2977-2991 (1996; Zbl 0866.12005)]. When calculated over the prime field, these Galois groups get enlarged into the semilinear groups \(\Gamma(m,q).\) Similarly, for any integer \(n>0,\) the Galois groups of the \(n\)-th iterates of these vectorial polynomials get enlarged from \(\text{GL}(m,q,n)\) to \(\Gamma(m,q,n),\) where \(\text{GL}(m,q,n)\) is the general linear group of the free module of rank \(m\) over the local ring \(\text{GF}(q)[T]/T^n\) and \(\Gamma(m,q,n)\) its semilinearization. Similar things hold when, instead of over rational function fields, the vectorials are considered over meromorphic function fields. A similar semilinear enlargement takes place when dealing with other classical groups, e.g. symplectic groups. The calculation of these various Galois groups leads to a determination of the algebraic closure of the ground fields in the splitting fields of the corresponding vectorial polynomials.NEWLINENEWLINEFor the entire collection see [Zbl 0941.00014].