On an infinite family of cubic fields with explicit fundamental units

Source abstract

For an integer $b\neq0,1$ let $θ$ be the unique real root of $f_b(x)=x^3-3bx-b^3$ and let $K_b=\mathbb{Q}(θ)$. We exhibit an explicit set of $b$ of positive density for which $η_b=-1/(θ-(b+1))$ is the fundamental unit of $K_b$, and we determine the set of $b$ for which $η_b$ is the square of a unit: it is parametrized by the Pell equation $D^2-3E^2=1$, hence infinite, and under an explicit mild condition on the field discriminant it accounts for all $b$ for which $η_b$ is not the fundamental unit. That condition fails for only four $b$ with $|b|\le3000$, and at one of them, $b=3$, the conclusion itself fails. In the order $\mathbb{Z}[η_b]$ generated by $η_b$, by contrast, $η_b$ is the fundamental unit for every $b$, so these exceptions are a phenomenon of the maximal order. As an application we construct infinitely many biquadratic fields whose $3$-class field tower has length greater than $1$.