We study the ramification properties of the extensions $\mathbb{Q}({\zeta }_m,\sqrt[m]{a})/\mathbb{Q}$ under the hypothesis that $m$ is odd and if $p\mid m$ than either $p\nmid v_p\left(a\right)$ or $p^{v_p\left(m\right)}\mid v_p\left(a\right)$ ($v_p\left(a\right)$ and $v_p\left(m\right)$ are the exponents with which $p$ divides $a$ and $m$). In particular we determine the higher ramification groups of the completed extensions and the Artin conductors of the characters of their Galois group. As an application, we give formulas for the $p$-adique valuation of the discriminant of the studied global extensions with $m=p^r$.

A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of $K^{\times }$ has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of $L^{\times }\setminus K^{\times }$. We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K.