TITLE: Cubic cyclic field families
QUESTION [0 upvotes]: Let $\mathbb{Q}$ is the field of rational numbers. If $f(t,x)=x^3+(t+2)x^2+(t-1)x-1$, show that for any integer $t$, adjoining a root $r$ of $f(t,x)$ to $\mathbb{Q}$ will yield a cyclic cubic field. Once this is shown, if possible, find a quintic (degree $5$) polynomial $f(t,x)$ (coefficients in terms of $t$) such that for any integer $t$, adjoining a root of $f(t,x)$ to $\mathbb{Q}$ will yield a cyclic quintic field. Thanks for help.

REPLY [1 votes]: Note that if (for fixed $t$) $\alpha$ is a root of $f(t, x)$ then so is $-1-\alpha^{-1}$. The transformation $y \mapsto -1 -y^{-1}$ has order three. Maybe it is possible to find something similar for the quintic case.
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