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It is known that the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$ is $\mathbf{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\cong\mathbb{Z}/n\mathbb{Z}$ and the splitting field of $\Phi_n(x)$ over $\mathbb{Q}$ is $\mathbf{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})\cong(\mathbb{Z}/n\mathbb{Z})^{\times}$. And $\mathbf{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})\cong(\mathbb{Z}/n\mathbb{Z})^{\times}$ is cyclic if and only if $n=2,4,p^k,2p^k$ where $p$ is an odd prime and $k\geq1$ is an integer, in this case $\Phi_n(x)$ has the cyclic group of order $\varphi(n)=1,2,p^{k-1}(p-1),p^{k-1}(p-1)$.

But for a fixed positive integer $n$, is there an explicit separable polynomial such that its Galois group over $\mathbb{Q}$ is cyclic of order $n$? Is there any reference about this problem? Thanks!

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    $\begingroup$ Find a prime p congruent to 1 mod n, and use the Z/nZ extension F/Q intermediate between Q and the n-th cyclotomic field Q(mu_p). To get an actual polynomial out of this, use the traces from Q(mu_p) to F of the nontrivial p-th roots of unity. $\endgroup$
    – Noam D. Elkies
    Commented Sep 11 at 4:37
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    $\begingroup$ You might be interested also in the theory of generic polynomials. See, for instance, the book "Generic polynomials: Constructive Aspects of the Inverse Galois Problem" (Jensen–Ledet–Yui, Cambridge University Press, 2003). $\endgroup$
    – Tom De Medts
    Commented Sep 11 at 12:14

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