Database of Local Fields

This site contains complete tables of low degree extensions of Q_p, for small p. It also allows you to enter a polynomial, and find the factors of p-adic algebra it defines in terms of entries from the tables, and to bound the Galois root discriminant of a global field.

Identify a p-adic algebra

Select the prime you want and enter your polynomial. The polynomial should be monic with integral coefficients, and in the variable x. Exponentiation is represented by ^; mulitiplication can be represented by * or left implicit. For example, you might enter:

x^4+3*x^2-x+1

x^4+3x^2-x+1

Compute a GRD

Enter a monic polynomial with integral coefficients in the variable x. Exponentiation is represented by ^; mulitiplication can be represented by * or left implicit. For example, you might enter:

x^4+3*x^2-x+1

x^4+3x^2-x+1

Tables of p-adic fields

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

= local root number

d = discriminant root field (see below for coding of quadratic extensions)

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

I = Inertia group

Wild Slopes = list of indices where the higher ramification groups change, with Artin's upper numbering. We exclude tame ramification (which would correspond to slope 1). A value is repeated m times if the order of the corresponding ramification group changes by a factor of p^j

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 below. Other subfields are given by their sample polynomials.

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

Quadratic Field Codes

Quadratic extensions of Q_p can be given by adjoining a square root of d. We give d with the additional convention that * represents a quadratic non-residue modulo p (if p is odd, * congruent to 5 modulo 8 for p = 2).

Old interface to tables of p-adic fields

Extensions of Q₂

Extensions of Q₃

Extensions of Q₅

Extensions of Q₇