A Database for Field Extensions of the Rationals

Abstract

This paper announces the creation of a database for number fields. It describes the contents and the methods of access, indicates the origin of the polynomials, and formulates the aims of this collection of fields.

A database of number fields

“…This database gives many complete determinations of K(G, D) in small degrees n, collecting previous results and going well beyond them. Our database complements the Klüners-Malle online database [25], which covers more groups and signatures, but is not as focused on completeness results and the behavior of primes. Like the Klüners-Malle database, our database is searchable and intralinked.…”

“…Degree 25 080 from PSL 2 (11). The only known field K 1 with Galois group PSL 2 (11) and root discriminant less than Ω first appeared in [25] and is the splitting field of…”

Artin L-functions of small conductor

“…Early work for quartics, quintics, sextics, and septics include respectively [2][3][4]7,13,[21][22][23]25,31]. Further results towards 2 in higher degrees are extractable from the websites associated to [10,12,14].…”

Multi-modular algorithm for computing the splitting field of a polynomial

“…Heuristic bound BH : We choose, as an heuristic, to begin the computation Proc CAND modulo the fifth power of the product of one, two or three minimal primes satisfying the condition state before. Comments on the experiments: We try our implementation on several polynomials given by [11] from degree 6 to 9 and some polynomials of greater degree (not more than 13) corresponding to interesting computation schemes. By using this heuristic bound, Proc CAND already computes in all the cases the final result, thus the remaining computation is the check procedure without any reconstruction.…”