The Radical Solution and Computational Complexity

TL;DR

This paper demonstrates that solving polynomials with rational coefficients is NP-hard and argues that P does not equal NP by relating graph isomorphisms and radical formulas, highlighting an impossible trinity in computational complexity.

Contribution

It establishes the NP-completeness of polynomial radical solutions and links graph isomorphism properties to the P versus NP problem, proposing a novel perspective.

Findings

Polynomial radical solving is NP-hard.

Graph isomorphism relates to radical formulas.

P does not equal NP based on these properties.

Abstract

The radical solution of polynomials with rational coefficients is a famous solved problem. This paper found that it is a $\mathbb{NP}$ problem. Furthermore, this paper found that arbitrary $ \mathscr{P} \in \mathbb{P}$ shall have a one-way running graph $G$, and have a corresponding $\mathscr{Q} \in \mathbb{NP}$ which have a two-way running graph $G'$, $G$ and $G'$ is isomorphic, i.e., $G'$ is combined by $G$ and its reverse $G^{-1}$. When $\mathscr{P}$ is an algorithm for solving polynomials, $G^{-1}$ is the radical formula. According to Galois' Theory, a general radical formula does not exist. Therefore, there exists an $\mathbb{NP}$, which does not have a general, deterministic and polynomial time-complexity algorithm, i.e., $\mathbb{P} \neq \mathbb{NP}$. Moreover, this paper pointed out that this theorem actually is an impossible trinity.