The Radical Solution and Computational Complexity
<|reference_start|>The Radical Solution and Computational Complexity: 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.<|reference_end|>
@article{zheng2024the,
    title={The Radical Solution and Computational Complexity},
    author={Bojin Zheng and Weiwu Wang},
    journal={arXiv preprint arXiv:2405.15790},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.15790},
    primaryClass={cs.CC}
}