Say $K$ is a degree $d$ number field, and let $L$ denote its normal closure. Using the primitive element theorem, we see that $K=\mathbb{Q}(\alpha)$, where $\alpha$ is of degree $d$. If $\alpha_{1},\dots,\alpha_{d}$ are its galois conjugates, we have $L=\mathbb{Q}(\alpha_{1},\dots,\alpha_{d})$ and consequently $[L:\mathbb{Q}]\leq d^{d}$.
My question is - is this really a sharp bound? Or can $[L:\mathbb{Q}]$ be bounded polynomially in $d$?
For example, if $K=\mathbb{Q}(2^{\frac{1}{d}})$ then $L=\mathbb{Q}(2^{\frac{1}{d}},\xi_{d})$ and so $L$ is of degree $d\cdot\psi(d)$ where $\psi$ is euler totient function, which is much smaller than $d^d$.
Thank you very much!