Algorithmic search for integer Abelian roots of a polynomial with integer Abelian coefficients

Abstract

In this work, we consider the operations over Abelian integers of rank $n$. By definition, such numbers are elements of the complex field and have the form of polynomials with integer coefficients from the $n$th degree primitive root of 1. In contrast, the degrees of such polynomials are not greater than Euler's totient function $\varphi(n)$. We provide an example to show that there are infinitely many Abelian integers inside any zero-centered circle on the complex plane. In this work, for considered operations we give in particular the algorithm of calculation of the inverse for the Abelian integer of rank $n$. It allows us to analyze not only the rings of such numbers but also the fields of Abelian integers. Natural arithmetics for such algebraic structures leads us to study  the polynomials with integer Abelian coefficients. Thus, in the presented work we also investigate the problem of finding roots of such polynomials. As a result, we provide an algorithm that finds the integer Abelian roots of the polynomials over the ring of Abelian integers. This algorithm is based on the proposed statement that all roots of the polynomial are bounded by some domain. The computer calculations confirm the statistical truth of the statement.

Keywords

• abelian number
• algorithm
• polynomial root