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

TL;DR

This paper develops an algorithm to determine the solvability of p-adic equations of the form x^q=a and applies it to classify 6-dimensional p-adic filiform Leibniz algebras, providing computational tools and new classifications.

Contribution

It introduces a solvability criteria algorithm for x^q=a over p-adic numbers and uses it to classify specific p-adic Leibniz algebras, a novel approach in this area.

Findings

Algorithm for solvability when q=p^m

Classification of 6-dimensional p-adic filiform Leibniz algebras for q=2,3,4,5,6

Computer program for criteria computation

Abstract

In this paper we study the $p$-adic equation $x^q=a$ over the field of $p$-adic numbers. We construct an algorithm of calculation of criteria of solvability in the case of $q=p^m$ and present a computer program to compute the criteria for fixed value of $m \leq p-1$. Moreover, using this solvability criteria for $q=2,3,4,5,6$, we classify $p$-adic 6-dimensional filiform Leibniz algebras.