Finding Solvable Fields

This paper presents a computational method that exploits the structure of cubic equations with totally real roots to systematically search for solvable number fields of higher prime degree. Using this approach, we construct totally real number fields with Galois groups F_5 F_7 ≅ C_7 ⋊ C_6 C_11 ⋊ C_5 F_11 ≅ C_11 ⋊ C_10 C_13 ⋊ C_3 C_13 ⋊ C_4 C_13 ⋊ C_6 F_13 ≅ C_13 ⋊ C_12 The paper contains 187 polynomials of prime degree (5, 7, 11 and 13), none of which appear in the LMFDB. Among them, 52 satisfy the strict single-prime ramification condition, while the remaining 135 are "clean" (no wild ramification). Complete radical solutions are provided for all of them. In total, approximately 200 new totally real solvable number fields not present in the LMFDB were discovered and submitted to the database. The remaining 13 fields are of non-prime degree (including degrees 9, 10, 15 and 16) and also satisfy single-prime ramification; they are not discussed in detail in this paper but have been included in the data submission.