All Questions

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{...

For a finite group G with a derived sequence of length 2, please tell me how to specifically construct a field that is a Galois extension of $\mathbb{Q}$ and whose Galois group is G. I made some edits ...

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 ...

Let $G$ be a finite group, $n=|G|$. Let $\rho:G\rightarrow GL(n,\mathbb{C})$ be the regular representation. Let $G \le H \le S_n$ be another group. Then we have $\mathbb{Q}[x_1,\cdots,x_n]^H \le \...

Suppose that we have Galois covering $$X\longrightarrow P^{1}_{\mathbb{Q}}$$ defined over the rationals which is cyclic, in the sense that the Galois group associated to the covering is a cyclic group....