Tables of Galois groups
This directory contains CSV files describing the Galois groups of polynomials over the p-adic field ℚp. Each file has d+2 columns, the first d+1 columns being the coefficients of a degree d polynomial, and the final column being the T-number (see here, here or here) of its Galois group. These were computed with the pAdicGaloisGroup package.
Unless otherwise specified in the notes column, the files contain precisely one defining polynomial for every field of the given degree.
Up to degree 12, the polynomials are the same as in the Local Fields Database (LFDB), which contains more information. The Galois groups were computed independently. These tables are marked “LFDB” in the notes column.
Occasionally we could not find the Galois group of all polynomials in a file. In such cases the T-number is recorded as 0 and the number of such ommissions is recorded in the notes column.
| p | d | File | Count | Notes |
|---|---|---|---|---|
| 2 | 2 | Download, Browse | 7 | LFDB. |
| 2 | 4 | Download, Browse | 59 | LFDB. |
| 2 | 6 | Download, Browse | 47 | LFDB. |
| 2 | 8 | Download, Browse | 1,823 | LFDB. |
| 2 | 10 | Download, Browse | 158 | LFDB. |
| 2 | 12 | Download, Browse | 5,493 | LFDB. |
| 2 | 14 | Download, Browse | 78 | All fields with ramification degree 2. |
| 2 | 14 | Download, Browse | 510 | All fields with ramification degree 14. |
| 2 | 16 | Download, Browse | 140 | All fields with ramification degree 4, singly ramified and elementary abelian over its maximal unramified subfield. Possibly several polynomials per field. |
| 2 | 18 | Download, Browse | 2,046 | All totally ramified fields. |
| 2 | 20 | Download, Browse | 511,318 | All totally ramified fields. Possibly several polynomials per field. |
| 2 | 22 | Download, Browse | 8,190 | All totally ramified fields. |
| 2 | 24 | Download, Browse | 8 | All fields with ramification degree 8, singly ramified and elementary abelian over its maximal unramified subfield. Possibly several polynomials per field. |
| 2 | 32 | Download, Browse | 120 | All fields with ramification degree 8, singly ramified and elementary abelian over its maximal unramified subfield. Possibly several polynomials per field. 3 ommissions. |
| 3 | 3 | Download, Browse | 10 | LFDB. |
| 3 | 6 | Download, Browse | 75 | LFDB. |
| 3 | 9 | Download, Browse | 795 | LFDB. |
| 3 | 12 | Download, Browse | 785 | LFDB. |
| 5 | 5 | Download, Browse | 26 | LFDB. |
| 5 | 10 | Download, Browse | 258 | LFDB. |