GAR-realization of classical groups
Let \(H\) be a finite nonabelian simple group. A regular Galois extension \(N/K (t)\) with Galois group \(G(N/K(t)) \simeq G = \Aut (H)\) is called a \(GA\)-realization of \(H\) over \(K\) if the fixed field \(N^H\) is a rational function field over \(K\). If further each field \(R/K (t)\) with \(R \overline K = N^H \overline K\) is a rational function field, then \(N/K(t)\) is called a \(GAR\)-realization of \(H\) over \(K\). We know that every embedding problem with kernel \(H^r\) has a solution over a Hilbertian field \(K\) if \(H\) possess a \(GAR\)-realization over \(K\). In this clearly written paper the author proves that certain infinite series of classical groups of Lie type have \(GA (R)\)-realizations over \(\mathbb{Q}\), for example 1) \(L_n (p)\) with \(n \equiv 1 \pmod 2\), \(p \geq 5\) a prime with \(p \not \equiv - 1 \pmod {12}\) and \(gcd (n,p - 1) = 1\); 2) certain infinite series of orthogonal groups \(O^\pm_{2n} (p)\); \(O_{2n + 1} (p)\). The proofs are based on rigidity methods of the constructive Galois theory and a detailed study of the groups under consideration.
scientific article; zbMATH DE number 861927
scientific article