﻿ Tables of Number Fields

#### In this page we provide the programs used to classify number fields of degree 8 and degree 9 with low discrminant, for the signatures (2,3), (4,2), (6,1), (1,4) and (3,3). We also present the data collected during the computations.

• If you need some recalls about Explicit Formulas and how to get lower bounds for discriminants: Recalls

• These are the gp files needed for the procedure to work:
nfdiscsearch8.gp
bounds_n8.gp
nfdiscsearch9.gp
bounds_n9.gp
In order to start the procedure, load the nfdiscsearch file corresponding to the desired degree: the corresponding bounds file will be automatically read.

• The following five links contain the polynomials found during the algorithmic procedure, plus the gp files containing the complete tables of polynomials with low discriminant in the corresponding signatures.
Every file collecting just polynomials is distinguished by the values of the corresponding parameters:

Degree n, which can be either 8 or 9.
Real embeddings r, which can be 2, 4 and 6 for degree 8 fields and 1, 3 for degree 9 fields.1
Value of the coefficient a1, which is an integer between 0 and 4.
Value of the coefficient a8 (a9), which for the collected lists can be either 1 or -1.
Parity value of the collected polynomials, which can be either odd or even.
• Degree 8, signature (2,3)

• Degree 8, signature (4,2)

• Degree 8, signature (6,1)

• Degree 9, signature (1,4)

• Degree 9, signature (3,3)

• A table of the classified number fields, with their discriminants, minimum polynomials and Galois groups, for every considered signature: here
The table contains:

• All the number fields of degree 8, signature (2,3) and |d_K|<= 5726301;

• All the number fields of degree 8, signature (4,2) and |d_K| <= 20829049;

• All the number fields of degree 8, signature (6,1) and |d_K| <= 79259702;

• All the number fields of degree 9, signature (1,4) and |d_K| <= 39657561;

• All the number fields of degree 9, signature (3,3) and |d_K| <= 146723910.

• A report on the time of computations for all the procedures: here
We finally give tables of number fields with signature (0,4), (8,0) and (9,0) up to certain discriminant bounds.
The upper bounds are the ones for which is known that the considered fields are the only ones: the references for the choice of the upper bounds and for the fields consist in the following papers.
• Diaz y Diaz, F. Petits discriminants des corps de nombres totalement imaginaires de degré 8. J. Number Theory 25 (1987), no. 1, 34–52.
• Pohst, M. Martinet J. Diaz y Diaz F. The minimum discriminant of totally real octic fields. J. Number Theory 36 (1990), no. 2, 145–159.
• Takeuchi, K. Totally real algebraic number fields of degree 9 with small discriminant. Saitama Math. J. 17 (1999), 63–85 (2000).
The tables are written in the format used by Megrez for number fields, presenting the discriminant, the coefficients of the minimum polynomial, the class number and the generators of the class group: