I already posted a question about a sum involving the degree of a Kummer extension.
Now I'm interested in a more specific fact about Kummer extensions.
From Hooley's paper "On Artin's conjecture", we know that if $k\_n=[\mathbb{Q}(\zeta\_n,a^{1/n}):\mathbb{Q}]$
is the degree of a Kummer extension for a fixed integer $a$, $a\neq 0,\pm1$; then writing $a=b^h$ for some integer $b=b\_0 b\_1^2$, with $b\_0$ squarefree, being $h$ the maximum possible exponent, we have
$$
k\_n=\frac{n\varphi(n)}{\delta(n)\gcd(n,h)}\;,
$$
where $\delta(n)=1,2$ depending on the congruence class of $a$ (mod 4).


My question: do we have a similar result in the case $a\in\mathbb{Q}$?
Do we have again an upper and lower bound of the type
$$
cn\varphi(n) \leq k\_n \leq C n\varphi(n) 
$$
for two fixed constants $c$ and $C$? Are those constants computable?


It's enough if any of you can suggest me the bibliography where I can see this case.


Thanks in advance. 



Bibliography suggestion for Kummer theory
$$\mathbb Q \left(\left( a \over b\right)^{1/n} \right)=\mathbb Q\left( \left( ab^{n-1}\right)^{1/n}\right)$$


