On the cyclotomic field $\mathbb{Q}(e^{2\pi {\bf i}/p})$ and Zhi-Wei Sun's conjecture on $\det M_p$

In 2019, Zhi-Wei Sun posed an interesting conjecture on certain determinants with Legendre symbol entries. In this paper, by using the arithmetic properties of $p$-th cyclotomic field and the finite field $\mathbb{F}_p$, we confirm this conjecture.