Examples of some \({\mathbb{Q}}\)-admissible groups
Recall that, if k is a number field, a finite group G is called k- admissible if G is isomorphic to Gal(K/k) for some number field K that is a maximal subfield of a finite-dimensional division algebra with center k. The authors prove the \({\mathbb{Q}}\)-admissibility of the groups \(A_ 6\), \(A_ 7\), \(\tilde A_ 7\), and thus complete the list of alternating groups which may be \({\mathbb{Q}}\)-admissible. The admissibility of the groups G in question is proved by exhibiting the fields K (for which \(G\cong Gal(K/ {\mathbb{Q}}))\) as the splitting fields of certain generalized Laguerre polynomials. The coefficients of these polynomials are constructed from expressions that are defined using certain rational points on the elliptic curves \(y^ 2=15x(x^ 2-4)\) and \(y^ 2=105x(x^ 2-4)\). The rational points, themselves, were found with the aid of a computer. One rather amusing fact pointed out by the authors is that the theory of heights shows that the numerator or denominator of the constant term of the polynomial for \(A_ 7\) has approximately \(1{1/2}\) billion digits.
scientific article; zbMATH DE number 4004310