Database of Local Fields

John W. Jones and David P. Roberts

This site contains complete tables of low degree extensions of the fields Q_p of p-adic numbers and also some interactive features. Details about the content and construction of the database are given in the following paper.

Some of the files underlying the database are also available at the bottom of this page.

Identify a p-adic algebra Q_p[x]/f (x)

Select the prime you want and enter your polynomial f(x). The polynomial should be monic, separable, with integral coefficients, and in the variable x. Exponentiation is represented by ^; multiplication can be represented by * or left implicit. For example, you might enter x^4+3*x^2-x+1 or equivalently x^4+3x^2-x+1 If the resulting algebra involves fields not contained in this database, you will be told that too.

Prime p:

Polynomial f(x):

Compute the Root Discriminant of a Galois Number Field

If all completions of Q[x]/f (x) are in the database, then bounds for the root discriminant of the splitting field of f (x) are computed. This root discriminant is referred to as the Galois root discriminant (GRD) of f, or the GRD of the algebra Q[x]/f (x).

Enter a monic polynomial with integral coefficients in the variable x as above.

Polynomial f(x):

Tables of p-adic fields

Prime:
Degree:

Currently supported tables are for primes p and degrees n where at least one of the following holds:

1. p and n are relatively prime
2. n = p
3. n < 12

Large tables (e.g., 2-adic octics or 3-adic nonics) may take a long time to load.

Format of the Tables

The tables are organized by the degree over Q_p and contain the following data:

c = p-adic valuation of the discriminant

e = ramification index

f = residue field degree

d   represents the discriminant root field, Q_p(sqrt(disc(f)), where f is any irreducible defining polynomial for the field. This field can be taken in the form Q_p(sqrt(d)) where we pick representatives d in Q_p^× modulo squares. Here, * = 5 if p = 2 and * is a quadratic non-residue if p is odd. Thus for p odd, the possibilities for d are 1, *, p, and p*, and for p=2, d is one of 1, *, 2, 2*, -1, -*, -2, or -2*.

= local root number

Polynomial = a sample defining polynomial. The polynomial is a link to the same polynomial in a form which can be readily selected and pasted into other programs.

G = Galois group, given in many cases in descriptive notation such as C_n for the cyclic group of order n, otherwise via its T number as in, for example, "The transitive groups of degree up to eleven", by Butler and McKay, Comm. Algebra 11, 1983. In all cases, Galois group names are links to pop-ups with group information.

I = Inertia group. These follow the same conventions as for Galois groups. In particular, they are clickable links to further information. In some cases, the inertia subgroup entry may only give partial information, such as the order of the inertia subgroup.

Slope Content gives information about the higher ramification filtration for the Galois group G, using Artin's upper numbering. It is explained in detail in the paper at the top of this page. In short, slope content takes the form [s_1, s_2, ..., s_j]^u_t. The s_i give indices where the wild higher ramification groups change counted with multiplicity, so that a value is repeated m times if the order of the corresponding ramification group changes by a factor of p^m. If there is no wild ramification, then no values are printed between the brackets. The values t and u give the tamely ramified and unramified degrees of the Galois closure respectively. We do not print t or u if its value is 1.

GMS = Galois Mean Slope, equal to the exponent of p in the root discriminant of the Galois closure of the field

Deg-j Subs = list of subfields of degree j over Q_p. Quadratic subfields are coded as listed above. Other subfields are given by their sample polynomials, with the exception of unramified subfields. Unramified degree n subfields are denoted simply by U_n.

Twin Algebra = a defining polynomial for the twin algebra of a sextic field. This twin polynomial is factored as a product of sample polynomials.

Files behind the web pages

We provide some of the files which are used by this site.

• Code
  • panayi.gp: a gp file with implimentations of Panayi's algorithm. If using this with other files, always load this first.
  • localgalois.gp: a gp file for computing Galois groups of local fields. In general, one needs some data precomputed for the field stored in the format used below for data files.
  • dynamic.gp: a gp file for generating dynamic parts of the database, e.g., lists of defining polynomials for unramified, tamely ramified, and degree p extensions of Q_p. It loads panayi.gp and localgalois.gp.
  • utilities.gp: a gp file for finding a polynomial in a database file. It assumes panayi.gp has been loaded.
• Data The following files are for cases where a prime p can be wildly ramified, and the degree is composite. The files themselves can be read into gp. Their format is described here. Files are separated by prime p and degree n over Q_p.
  • p=2 n=4
  • p=2 n=6
  • p=2 n=8
  • p=2 n=10
  • p=3 n=6
  • p=3 n=9
  • p=5 n=10