The Radical Solution and Computational Complexity

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 $\mathcal{P}\in \mathbb{P}$ shall have a one-way running graph $G$, and have a corresponding $\mathcal{Q}\in \mathbb{NP}$ which have a two-way running graph $G^{\prime }$, $G$ and $G^{\prime }$ is isomorphic, i.e., $G^{\prime }$ is combined by $G$ and its reverse $G^{-1}$. When $\mathcal{P}$ 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., $\mathbb{P}\ne \mathbb{NP}$. Moreover, this paper pointed out that this theorem actually is an impossible trinity.