Septic Number Fields Which are Ramified Only at One Small Prime

AbstractWe find all number fields which can be generated from a degree 7 polynomial satisfying the conditions that only one prime, p, ramifies and p< 11

On number fields with equivalent integral trace forms

Let $K$ be a number field. The \textit{integral trace form} is the integral quadratic form given by $\text{tr}_{K/\mathbb{Q}}(x^2)|_{O_{K}}.$ In this article we study the existence of non-conjugated number fields with equivalent integral trace forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral trace forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields of fundamental discriminant have equivalent trace zero forms if and only if they are conjugated

Mixed Degree Number Field Computations

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 25 octic fields with Galois group PSL₂(7) and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group 2³:GL₃(2) of smallest absolute discriminant

A database of number fields

We describe an online database of number fields which accompanies this paper The database centers on complete lists of number fields with prescribed invariants. Our description here focuses on summarizing tables and connections to theoretical issues of current interest.Comment: 25 pages, 1 figur

Number fields unramified away from 2

We consider finite extensions of the rationals which are unramified except for at 2 and infinity. We show there are no such extensions of degrees 9 through 15

Number Fields Ramified at One Prime

For G a finite group and p a prime, a G-p field is a Galois number field K with Gal(K/Q)≅G and disc(K)=±pa for some a. We study the existence of G-p fields for fixed G and varying p