What does this research mean for the field?

The fourth graded layer of the Galois group of the maximal pro-5 extension of Q(ζ₅) unramified outside places above 5 and 11 has dimension 510, and its degree-four relations are determined entirely by quadratic relations without dependence on the Frobenius longitude. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To determine the algebraic structure and dimensions of the fourth graded layer and the quotient group for the maximal pro-5 Galois group unramified outside 5 and 11.

August 30, 2026Open Access

A Galois Group of Order 5^649: The Fourth Graded Layer Is Longitude-Blind

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To determine the algebraic structure and dimensions of the fourth graded layer and the quotient group for the maximal pro-5 Galois group unramified outside 5 and 11.
Carried out exact machine-verified integer linear algebra to construct the 234 explicit degree-four relator matrix columns within the 728-dimensional free module.
Analyzed the semisimple action and isotypic decompositions of the base Galois group on the relator ideal, quotient, and free group layer.
Computed and compared the invariant dimensions and ideal intersections against an independent prime branch at q = 61.
Determined that the fourth graded piece gr₄(G) has dimension 510, producing a quotient G/λ₅(G) of order 5⁶⁴⁹ (a 454-digit integer) with a relator ideal of rank 218 inside the 728-dimensional free layer.
Proved a longitude-blindness theorem where the degree-four columns derive entirely from the four quadratic relation rows, showing no new arithmetic enters from the Frobenius longitude due to vanishing cubic syzygies.
Identified that the Galois action splits into isotypic multiplicities of (54, 55, 55, 54) on the ideal and (125, 131, 131, 123) on the quotient, with an exact 16-dimensional intersection between the ideals at q = 11 and q = 61.

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Implication

Algebraic study demonstrates that the fourth graded layer of a pro-5 Galois group has order 5^649 and is longitude-blind, revealing that degree-four relations depend purely on quadratic terms.

Cite This Study

xuewen guan (2026) studied this question.

synapsesocial.com/papers/6a93f1396c1a8fb52e79e06f https://doi.org/10.5281/zenodo.22162285

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Implication

Algebraic study demonstrates that the fourth graded layer of a pro-5 Galois group has order 5^649 and is longitude-blind, revealing that degree-four relations depend purely on quadratic terms.

A Galois Group of Order 5^649: The Fourth Graded Layer Is Longitude-Blind

Key Points

Discussion

Implication

Cite This Study

A Galois Group of Order 5^649: The Fourth Graded Layer Is Longitude-Blind

Key Points

Discussion

Implication

Cite This Study

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