Mixed degree number field computations

Abstract

We present a method for computing complete lists of number fields in cases where the Galois group, as an abstract group, appears as a Galois group in smaller degree. We apply this method to find the twenty-five octic fields with Galois group PSL 2 (7) and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group 2 3 : GL 3 (2) of smallest absolute discriminant.

Artin L-functions of small conductor

“…It is prohibitive to compute the first octic discriminant by searching among octic polynomials. In [9] we carried out a long search of septic polynomials, examining all local possibilities giving an octic discriminant at most 30. This computation shows that |L(GL 3 (2), χ 7 ; 48.76)| = 25 and in particular identifies δ 1 = 21 8/7 ≈ 32.44.…”