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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.

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

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