On an infinite family of cubic fields with explicit fundamental units
Abstract
For an integer b≠0,1 let θ be the unique real root of fb(x)=x3−3bx−b3 and let Kb=ℚ(θ). We exhibit an explicit set of b of positive density for which ηb=−1/(θ−(b+1)) is the fundamental unit of Kb, and we determine the set of b for which ηb is the square of a unit: it is parametrized by the Pell equation D2−3E2=1, hence infinite, and under an explicit mild condition on the field discriminant it accounts for all b for which ηb is not the fundamental unit. That condition fails for only four b with |b|≤3000, and at one of them, b=3, the conclusion itself fails. In the order ℤ[ηb] generated by ηb, by contrast, ηb 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 $1$.
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BibTeX
@misc{kimura2026infinite,
title = {{On an infinite family of cubic fields with explicit fundamental units}},
author = {Iwao Kimura and Hikaru Umemoto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.37389},
eprint = {2609.37389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37389},
}
RIS
TY - UNPB
TI - On an infinite family of cubic fields with explicit fundamental units
AU - Iwao Kimura
AU - Hikaru Umemoto
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.37389
UR - https://arxiv.org/abs/2609.37389
AB - For an integer b≠0,1 let θ be the unique real root of f_b(x)=x^3−3bx−b^3 and let K_b=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|≤3000, and at one of them, b=3, the conclusion itself fails. In the order 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 $1$.
ER -