Automorphisms of congruence function fields
Let \(k\) be a finite field. For a function field \(K\) over \(k\) and \(m\geq 3\), it is proven that there are infinitely many non-isomorphic function fields \(L\) such that \(L/K\) is a separable extension of degree \(m\) and \(Aut_k L=\{\text{Id}\}\). It is also shown that for a finite group \(G\), there are infinitely many non-isomorphic function fields \(L/k\) such that \(Aut_k L\cong G\). Finally, given any finite nilpotent group \(G\) such that \(| G| >1\) and (\(| G|, | k| -1)=1\), and any function field \(K\) over \(k\), there are infinitely many non-isomorphic function fields \(L\) over \(k\) with \(\mathrm{Gal}(L/K) = \Aut_k L\cong G\).
scientific article; zbMATH DE number 4136072