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Let $\mathbb{G}= \{g_1,\dots,g_n\}$ be a finite group and $\rho$ its regular representation. Let $x_1,\dots,x_n$ be indeterminates and let $x = (x_1,\dots,x_n)^\top$. Let the matrix $G$ be defined ...

It is known that the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$ is $\mathbf{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\cong\mathbb{Z}/n\mathbb{Z}$ and the splitting field of $\Phi_n(x)$ over $\mathbb{...