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October 13, 2025Open Access

Greenberg's conjecture for real quadratic number fields

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The orders of the 3-class groups remain bounded as n increases, consistent with Greenberg's conjecture.
This research computes the 3-class groups for real quadratic fields with discriminants less than 100,000.
The analysis focuses on the cyclotomic extensions of the real quadratic fields.
Findings support the conjecture's validity, providing insights into the structure of these number fields.

Implication

Analysis confirms Greenberg's conjecture on 3-class groups in real quadratic fields, suggesting stability in their behavior across extensions.

Abstract

We compute the $3$-class groups Aₙ of the fields Fₙ in the cyclotomic Z₃-extensions of the real quadratic fields of discriminant $f<100,000$. In all cases the orders of Aₙ remain bounded as n goes to infinity. This is in agreement with Greenberg's conjecture.

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Cite This Study

Mercuri et al. (2025) studied this question.

synapsesocial.com/papers/68ecc715d1cc7436f7d18bcf https://doi.org/10.48550/arxiv.2503.00819

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Implication

Analysis confirms Greenberg's conjecture on 3-class groups in real quadratic fields, suggesting stability in their behavior across extensions.

Greenberg's conjecture for real quadratic number fields

Key Points

Implication

Abstract

Discussion

Cite This Study

Greenberg's conjecture for real quadratic number fields

Key Points

Implication

Abstract

Discussion

Cite This Study

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