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A characterization of Eisenstein polynomials generating extensions of degree p 2 and cyclic of degree p 3 over an unramified 𝔭 -adic field

Maurizio Monge (2014)

Journal de Théorie des Nombres de Bordeaux

Let p ≠ 2 be a prime. We derive a technique based on local class field theory and on the expansions of certain resultants allowing to recover very easily Lbekkouri’s characterization of Eisenstein polynomials generating cyclic wild extensions of degree p 2 over ℚ p , and extend it to when the base fields K is an unramified extension of ℚ p .When a polynomial satisfies a subset of such conditions the first unsatisfied condition characterizes the Galois group of the normal closure. We derive a complete classification...

A fast algorithm for polynomial factorization over ℚ p

David Ford, Sebastian Pauli, Xavier-François Roblot (2002)

Journal de théorie des nombres de Bordeaux

We present an algorithm that returns a proper factor of a polynomial Φ ( x ) over the p -adic integers ℤ p (if Φ ( x ) is reducible over ℚ p ) or returns a power basis of the ring of integers of ℚ p [ x ] / Φ ( x ) ℚ p [ x ] (if Φ ( x ) is irreducible over ℚ p ). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm.

Beyond two criteria for supersingularity: coefficients of division polynomials

Christophe Debry (2014)

Journal de Théorie des Nombres de Bordeaux

Let f ( x ) be a cubic, monic and separable polynomial over a field of characteristic p ≥ 3 and let E be the elliptic curve given by y 2 = f ( x ) . In this paper we prove that the coefficient at x 1 2 p ( p - 1 ) in the p –th division polynomial of E equals the coefficient at x p - 1 in f ( x ) 1 2 ( p - 1 ) . For elliptic curves over a finite field of characteristic p , the first coefficient is zero if and only if E is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the zero loci...

Classification of p-adic 6-dimensional filiform Leibniz algebras by solutions of x q = a

Manuel Ladra, Bakhrom Omirov, Utkir Rozikov (2013)

Open Mathematics

We study the p-adic equation x q = a over the field of p-adic numbers. We construct an algorithm which gives a solvability criteria in the case of q = p m and present a computer program to compute the criteria for any fixed value of m ≤ p − 1. Moreover, using this solvability criteria for q = 2; 3; 4; 5; 6, we classify p-adic 6-dimensional filiform Leibniz algebras.

Computing r -removed P -orderings and P -orderings of order h

Keith Johnson (2010)

Actes des rencontres du CIRM

We develop a recursive method for computing the r -removed P -orderings and P -orderings of order h , the characteristic sequences associated to these and limits associated to these sequences for subsets S of a Dedekind domain D . This method is applied to compute these objects for S = ℤ and S = ℤ ∖ p ℤ .

Dynamique des polynômes quadratiques sur les corps locaux

Robert Benedetto, Jean-Yves Briend, Hervé Perdry (2007)

Journal de Théorie des Nombres de Bordeaux

Dans cette note, nous montrons que la dynamique d’un polynôme quadratique sur un corps local peut être déterminée en temps fini, et que l’on a l’alternative suivante : soit l’ensemble de Julia est vide, soit P y est conjugué au décalage unilatéral sur 2 symboles.

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