REVIEW 2 minor 6 references
On Euclidean systems of ray classes
T0 review · 0 major / 2 minor · reviewed 2026-07-03 · grok-4.3
Pith's one-line read Assuming GRH, every generating set of the ray class group with modulus a power of an odd prime is a Euclidean system of ray classes, for totally real Galois fields of degree at least 3 where the prime does not split completely.
desk verdict Extends Euclidean systems to ray classes with an unconditional generation result and a GRH-conditional claim that generating sets are Euclidean in certain fields. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Euclidean system of ray classes: a finite set of representatives in the ring of integers that permits a Euclidean division algorithm with respect to the action of the ray class group modulo the given conductor.
What would settle it
An explicit totally real Galois field K of degree at least 3, an odd prime p that does not split completely, some N, and a generating set S of Cl_K^{(p)^N} such that S fails the Euclidean division property for at least one pair of elements.
Extended reading notes
Core claim
We formulate the notion of a Euclidean system of ray classes and prove that every such system generates the corresponding ray class group. Assuming GRH, if K is a totally real Galois number field of degree n≥3 and p is an odd rational prime that does not split completely in K, then for every N>0 every generating set of the ray class group Cl_K^{(p)^N} with modulus (p)^N is a Euclidean system.
Load-bearing premise
The generalized Riemann hypothesis must hold for the claim that every generating set is Euclidean; the generation property itself relies only on the new definition.
Editorial extensions
If this is right
- Every Euclidean system of ray classes generates the ray class group.
- The Euclidean property holds for all generators once GRH is assumed in the stated setting.
- The result applies uniformly for every exponent N on the prime power modulus.
Reading between the lines
- The conditional result may supply explicit generators that can be used to compute ray class groups via a Euclidean algorithm.
- It raises the question whether similar unconditional statements hold when the degree is 2 or when the prime splits completely.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces Euclidean systems of ray classes, extending Lenstra's Euclidean ideal classes and Treatman's Euclidean systems. It proves unconditionally that every Euclidean system generates the corresponding ray class group. Assuming GRH, for a totally real Galois number field K of degree n≥3 and odd prime p not splitting completely in K, every generating set of the ray class group Cl_K^{(p)^N} (modulus (p)^N) is a Euclidean system, for any N>0.
Significance. If the results hold, the work provides a conditional characterization of generating sets as Euclidean systems under GRH for specified fields and primes, extending prior notions to ray classes. The unconditional generation result follows directly from the definition and is a basic consistency check. The GRH-conditional statement offers a strong, falsifiable claim in the context of class field theory and Euclidean algorithms.
minor comments (2)
- Abstract contains typographical spacing errors: 'corre sponding' and 'ratio nal'.
- The manuscript should include a brief comparison of the new ray-class definition with Treatman's original Euclidean systems to clarify the extension.
Simulated Author's Rebuttal
We thank the referee for the positive summary and recommendation of minor revision. The report correctly identifies the main results: the unconditional generation property and the GRH-conditional characterization for the specified fields. No specific major comments were provided in the report, so we have no individual points requiring response or revision at this stage.
Circularity Check
No significant circularity identified
full rationale
The paper formulates a new definition of Euclidean systems of ray classes, then proves the basic property that any such system generates the ray class group (a direct consequence of the definition, not a reduction to inputs by construction). The main theorem is a conditional result under GRH showing that generating sets satisfy the Euclidean property for specified fields; this is an independent derivation relying on GRH rather than self-definition, fitted parameters, or load-bearing self-citations. No steps match the enumerated circularity patterns.
Assumptions & free parameters
assumptions (1)
- domain assumption Generalized Riemann Hypothesis (GRH)
Cite this review
Pith. "Pith review of On Euclidean systems of ray classes." pith.science (2026). https://pith.science/paper/DT6P2SF4
@misc{pith2026260701703,
author = {Pith},
title = {Pith review of: On Euclidean systems of ray classes},
year = {2026},
howpublished = {\url{https://pith.science/paper/DT6P2SF4}},
note = {Machine review of arXiv:2607.01703}
}
abstract
Lenstra introduced the notion of Euclidean ideal classes, and Treatman extended it to Euclidean systems. In this paper, we formulate Euclidean systems for ray classes, and study their basic properties. In particular, we show that every Euclidean system of ray classes generates the corre sponding ray class group. We further prove, assuming GRH, that if $K$ is a totally real Galois number field of degree $n\ge 3$ and $p$ is an odd ratio nal prime which does not split completely in $K$, then for every $N>0$, every generating set of the ray class group $Cl_K^{(p)^N}$ with modulus $(p)^N$ is a Euclidean system.
Reference graph
Works this paper leans on
Reviewed July 3, 2026 · model on record in the stance chip above.
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