Tenth degree number fields with quintic fields having one real place

Abstract:

In this paper, we enumerate all number fields of degree $10$ of discriminant smaller than $10^{11}$ in absolute value containing a quintic field having one real place. For each one of the $21509$ (resp. $18167)$ found fields of signature $(0,5)$ (resp. $(2,4))$ the field discriminant, the quintic field discriminant, a polynomial defining the relative quadratic extension, the corresponding relative discriminant, the corresponding polynomial over $\mathbb {Q}$, and the Galois group of the Galois closure are given. In a supplementary section, we give the first coincidence of discriminant of $19$ (resp. $20)$ nonisomorphic fields of signature $(0,5)$ (resp. $(2,4))$.