integer $M$, we denote by $E[M]$ the group of $M$-torsion points of $E$, that is, the set $\{P \in E({\overline{K}}) : MP=0\}$ equipped with the group law inherited from $E$. Moreover, we denote by $K_M$ the *$M$-th torsion field* $K(E[M])$ of $E$, namely, the finite extension of $K$ obtained by adjoining the coordinates of all the $M$-torsion points of $E$. For each positive integer $N$ dividing $M$, we let $N^{-1}\alpha:={\left\{ \beta\in E({\overline{K}})\mid N\beta=\alpha \right\}}$ denote the set of $N$-division points of $\alpha$ and set $$\begin{aligned}
K_{M,N}:=K(E[M],N^{-1}\alpha).\end{aligned}$$ The field $K_{M,N}$ is called the *$(M,N)$-Kummer extension* of $K$ (related to $\alpha$), and both $K_M$ and $K_{M,N}$ are finite Galois extensions of $K$.

It is a classical question to study the degree of $K_{M,N}$ over $K_M$ as $M,N$ vary, see for example [@Durham Théorème 5.2], [@Hindry Lemme 14], or Ribet’s foundational paper [@MR552524]. In particular, it is known that there exists an integer $C=C(E/K, \alpha)$, depending only on $E/K$ and $\alpha$, such that $$\begin{aligned}
\frac{N^2}{\left[K_{M,N}:K_M\right]}\quad\text{divides}\quad C\end{aligned}$$ for every pair of positive integers $(M,N)$ with $N\mid M$.

The aim of this paper is to give an explicit version of this result, and to show that it can even be made uniform when the base 