It's known that all abelian groups are regularly realizable over $\mathbb{Q}(x)$, but it occurred to me that I don't even have an example of a cyclic regular extension of $\mathbb{Q}(x)$ handy.


So: what is an example of a regular realization of $C\_5$ over $\mathbb{Q}(x)$?



What is an example of a regular realization of $C_5$ over $\mathbb{Q}(x)$?
Emma Lehmer's quintic
\begin{align\*}
y^5 +& x^2y^4 - (2x^3 + 6x^2 + 10x + 10)y^3 +\\
 &(x^4 + 5x^3 + 11x^2 + 15x + 5)y^2 + (x^3 + 4x^2 +10x + 10)y + 1
\end{align\*}
has Galois group $C\_5$ over ${\mathbf Q}(x)$ and the splitting field over ${\mathbf Q}(x)$ is a regular extension. Her paper is "Connection between Gaussian periods and cyclic units" Math. Comp. 50 (1988), 535--541.


Typing "Lehmer quintic" in Google produces several hits which will give more information.


