Computing the Galois group of a polynomial over a p-adic field

Computing the Galois group of a polynomial

This article outlines techniques for computing the Galois group of a polynomial over the rationals, an important operation in computational algebraic number theory. In particular, the linear resolvent polynomial method of [6] will be described.

Group Rings over the p-Adic Integers

On a recursive equation over a p-adic field

In the paper we completely describe the set of all solutions of a recursive equation, arising from the Bethe lattice models over p-adic numbers. Mathematics Subject Classification: 46S10, 12J12.

TORSION p-ADIC GALOIS REPRESENTATIONS

Let p be a prime, K a finite extension of Qp and T a finite free Zp-representation of Gal(K̄/K). We prove that T ⊗Zp Qp is semi-stable (resp., crystalline) with Hodge-Tate weights in {0, . . . , r} if and only if for all n, T/pnT is torsion semi-stable (resp., crystalline) with Hodge-Tate weights in {0, . . . , r}. Résumé. (Représentations galoisiennnes p-adiques de torsion) Soient p un nombre p...

Computing the $p$-adic Canonical Quadratic Form in Polynomial Time

An n-ary integral quadratic form is a formal expression Q(x1, · · · , xn) = ∑ 1≤i,j≤n aijxixj in nvariables x1, · · · , xn, where aij = aji ∈ Z. We present a randomized polynomial time algorithm that given a quadratic form Q(x1, · · · , xn), a prime p, and a positive integer k outputs a U ∈ GLn(Z/p Z) such that U transforms Q to its p-adic canonical form.