The construction of SL(2,3)-polynomials
In this paper polynomials over \({\mathbb{Z}}\) with Galois group the special linear group SL(2,3) are constructed. Since SL(2,3) is the nontrivial extension of the alternating group \(A_ 4\) with kernel \({\mathbb{Z}}/2{\mathbb{Z}}\), the realization of SL(2,3) as a Galois group over \({\mathbb{Q}}\) can be considered as an embedding problem. Let f(X)\(\in {\mathbb{Q}}[X]\) be a quartic polynomial with Galois group \(A_ 4\). The authors prove that, under some conditions on f, the Galois group of \(F(X)=f(X^ 2)\) is isomorphic either to SL(2,3) or \(C_ 2\times A_ 4\), or a certain group of order 96. Next, they give a computational criterion to decide whether this Galois group is isomorphic to SL(2,3). Totally real quartic polynomials f(X) over \({\mathbb{Q}}\) with Galois group \(A_ 4\) are constructed and for some of them it is checked that the Galois group of \(F(X)=f(X^ 2)\) is SL(2,3).
scientific article; zbMATH DE number 3902774