I tried to solve the following problem, but I couldn't; I hope can you help me.
Let $G$ the matrix group $G=\left.\left\{ \left(\begin{array}{cc} a & b \\ 0 & 1 \end{array}\right) \;\right|\; a,b \in \mathbb{F}_p, a \neq 0 \right\},$ with $p$ prime. Build a Galois extension $K$ of $\mathbb{Q}$ such that $\mathrm{Gal}(K/\mathbb{Q}) \cong G$. In addition, , determine the intermediate extension for the subgroups $H_1=\left.\left\{ \left(\begin{array}{cc} a & 0 \\ 0 & 1 \end{array}\right) \;\right|\; a \in \mathbb{F}_p, a \neq 0 \right\},$ and $H_2=\left.\left\{ \left(\begin{array}{cc} 1 & b \\ 0 & 1 \end{array}\right) \;\right|\; b \in \mathbb{F}_p, a \neq 0 \right\}.$
Thanks for you answer.