Let us consider the absolute Galois group ${\mathrm{Gal}}(\overline{{\Bbb Q}_p}/{\Bbb Q}_p)$ of the local field ${\Bbb Q}_p$. It is widely-known that ${\mathrm{Gal}}(\overline{{\Bbb Q}_p}/{\Bbb Q}_p)$ is topologically generated by two elements. That is, we have a certain di-generated discrete subgroup $L_p < {\mathrm{Gal}}(\overline{{\Bbb Q}_p}/{\Bbb Q}_p)$ such that the profinite completion of $L_p$ becomes ${\mathrm{Gal}}(\overline{{\Bbb Q}_p}/{\Bbb Q}_p)$.
Q. Is $L_p$ indepenedent of $p$? I.e., does it hold that $L_p \cong L_q$ for $p \not=q$?