Let $f \in \mathbb{Z}[x]$ be a separable monic polynomial, with $f(0) \neq 0$, and $p$ be a prime number. Also, let $L$ be the splitting field of $f$ over $\mathbb{Q}_p$ and let $a_1, \ldots, a_n \in L$ be all the roots of $f$. Finally, let $b_1, \ldots, b_n \in \mathbb{C}$ also be the roots of $f$, but this time taken in the complex numbers.
Is is true that the multiplicative group generated by $a_1, \ldots, a_n$ is isomorphic to the multiplicative group generated by $b_1, \ldots, b_n$?
I would say YES, since the splitting field is unique up to isomorphism, but $L$ is a splitting field over $\mathbb{Q}_p$, while the roots in $\mathbb{C}$ are considered as in a splitting field over $\mathbb{Q}$. So I'm not convinced...
Thank you.
Well, since $f\in\Bbb Q[x]$, all its roots $\{\rho_i\}$ are algebraic over $\Bbb Q$. Call the group generated by these $G$. It’s a finitely generated abelian group that is injected both into an algebraic closure of $\Bbb Q_p$ and into $\Bbb C$. What the splitting fields of $f$ are, as subfields of the nonarchimedean and the archimedean algebraically closed fields, is immaterial, I believe.