An Algorithm for Solving Solvable Polynomial Equations of Arbitrary Degree by Radicals

TL;DR

This paper introduces an algorithm for solving solvable polynomial equations of any degree by radicals, relying on known Galois groups and approximate roots to efficiently compute exact radical solutions.

Contribution

The work presents a novel algorithm that computes radical roots of solvable polynomials of arbitrary degree using Galois group information and approximate roots, without reducing symmetric polynomials.

Findings

Algorithm complexity is roughly proportional to the fourth power of the Galois group's size.

Exact radical roots can be obtained without handling large polynomials or symmetric polynomial reduction.

Approximate roots can be used to reduce computational effort.

Abstract

This work provides a method(an algorithm) for solving the solvable unary algebraic equation $f(x)=0$ ($f(x)\in\mathbb{Q}[x]$) of arbitrary degree and obtaining the exact radical roots. This method requires that we know the Galois group as the permutation group of the roots of $f(x)$ and the approximate roots with sufficient precision beforehand. Of course, the approximate roots are not necessary but can help reduce the quantity of computation. The algorithm complexity is approximately proportional to the 4th power of the size of the Galois group of $f(x)$. The whole algorithm doesn't need to deal with tremendous polynomials or reduce symmetric polynomials.