What question did this study set out to answer?

Establish an explicit criterion for the 8-rank (r₈ = 1) of narrow class groups in real quadratic fields with fundamental discriminant D = 4AB using Rédei matrices, Artin–Rédei symbols, and continued fractions.

August 16, 2026Open Access

Explicit 8-Rank Criteria in a Squarefree Composite Family: Rédei Kernels, Artin Symbols, and Continued-Fraction Centers

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Authors

KMKenshirou MoriwakiHokkaido Research Organization

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Implication

Computational algebraic analysis demonstrates explicit 8-rank criteria for real quadratic fields using Rédei matrices and continued fractions, enabling exact narrow class group classification.

Key Points

Establish an explicit criterion for the 8-rank (r₈ = 1) of narrow class groups in real quadratic fields with fundamental discriminant D = 4AB using Rédei matrices, Artin–Rédei symbols, and continued fractions.
Derived a four-bit formula for column-normalized Rédei matrices and an explicit basis for right kernels when the 4-rank equals one.
Connected right-kernel membership to Stevenhagen's Artin–Rédei symbol admissibility and the central complete quotient of the continued fraction of √(AB).
Executed programmatic verification scripts in PARI/GP and Python across comprehensive finite populations to cross-check class-field invariants and twist invariance.
Demonstrated that the 8-rank criterion reduces to evaluating a single Artin–Rédei symbol determined by the central quotient and ramified-prime support.
Confirmed the Artin criterion and twist invariance across 194 out of 194 test cases.
Verified the unified Rédei matrix across 19,927 inputs and the right-kernel formula on all 10,098 cases with r₄ = 1, covering complete class-number datasets of 4,440 and 5,658 cases across residue layers.

Cite This Study

Kenshirou Moriwaki (2026) studied this question.

synapsesocial.com/papers/6a817aa1f2fb91fc834ae9ae https://doi.org/10.5281/zenodo.21950744

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Implication

Computational algebraic analysis demonstrates explicit 8-rank criteria for real quadratic fields using Rédei matrices and continued fractions, enabling exact narrow class group classification.

Explicit 8-Rank Criteria in a Squarefree Composite Family: Rédei Kernels, Artin Symbols, and Continued-Fraction Centers

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Cite This Study

Explicit 8-Rank Criteria in a Squarefree Composite Family: Rédei Kernels, Artin Symbols, and Continued-Fraction Centers

Discussion

Member takes

Implication

Key Points

Cite This Study

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