pith. sign in

arxiv: 2606.10521 · v1 · pith:LMLERR45new · submitted 2026-06-09 · 🧮 math.NT · math.AC

Unique decomposition of orders

Gaurav Digambar Patil This is my paper

Pith reviewed 2026-06-27 11:58 UTC · model grok-4.3

classification 🧮 math.NT math.AC
keywords ordersnumber fieldsirreducible ordersconductorindexunique decompositionFurtwangler criterion
0
0 comments X

The pith

Any order in a number field decomposes uniquely as an intersection of irreducible orders.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a unique decomposition for orders inside number fields. Every such order equals the intersection of irreducible orders in exactly one way. Under this splitting the index of the order inside the ring of integers multiplies across the factors, while the conductor breaks into pairwise coprime ideals. The same decomposition yields a strengthened form of the Furtwangler criterion that classifies conductors of orders over the integers and settles many related structural questions.

Core claim

We establish a Fundamental Theorem of Orders (FTO), which allows us to express any order (in a number field) uniquely as an intersection of irreducible orders. Along this decomposition, the index (in the ring of integers) distributes multiplicatively, and the conductor factors into pairwise co-prime ideals. We use it to show a more general version of Furtwangler criterion about the structure of conductors of orders over Z, as this answers a wide variations of such questions.

What carries the argument

The Fundamental Theorem of Orders, a unique decomposition of any order as an intersection of irreducible orders that makes index multiplicative and conductor coprime-factorable.

If this is right

  • The index of any order in the ring of integers equals the product of the indices of its irreducible factors.
  • The conductor of the order factors into pairwise coprime ideals, one for each irreducible component.
  • A general version of the Furtwangler criterion classifies the possible conductors of orders over the integers.
  • Many structural questions about conductors of orders receive uniform answers through the same decomposition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The decomposition may reduce the computational cost of enumerating orders inside a fixed number field by factoring the problem into irreducible pieces.
  • It supplies a canonical way to compare orders across different number fields that share the same conductor factors.
  • One could test whether the same uniqueness holds when orders are replaced by more general Z-lattices inside the field.

Load-bearing premise

The intersection of any two orders is again an order and both the conductor and the index behave well under intersections, without further restrictions on the number field.

What would settle it

An explicit order inside a number field whose representation as an intersection of irreducible orders is either non-unique or fails to make the index multiplicative or the conductor ideals pairwise coprime.

read the original abstract

We establish a Fundamental Theorem of Orders (FTO), which allows us to express any order (in a number field) uniquely as an intersection of \textit{irreducible orders}. Along this decomposition, the index (in the ring of integers) distributes multiplicatively, and the conductor factors into pairwise co-prime ideals. We use it to show a more general version of Furtwangler criterion about the structure of conductors of orders over $\Z$, as this answers a wide variations of such questions. In a future work, we will also give applications to weighted enumeration of number fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript establishes a Fundamental Theorem of Orders (FTO) asserting that every order O in a number field K can be expressed uniquely as an intersection of irreducible orders. Under this decomposition the index [O_K : O] factors multiplicatively and the conductor of O factors into pairwise coprime ideals of O_K. The theorem is applied to obtain a generalization of the Furtwängler criterion describing the structure of conductors of orders over Z; applications to weighted enumeration of number fields are deferred to future work.

Significance. If the claimed decomposition and its arithmetic consequences hold, the result supplies a canonical factorization of orders that is compatible with the standard multiplicative behavior of indices and conductors when the latter are coprime. Such a theorem would organize the lattice of orders in a number field and could streamline arguments that previously relied on ad-hoc choices of orders or conductors.

major comments (2)
  1. The manuscript states the FTO and the multiplicative properties but supplies neither a definition of 'irreducible order' nor a proof of existence or uniqueness of the decomposition. Without these, the central claim cannot be verified.
  2. No explicit statement is given of the ambient number field (e.g., whether it is required to be Galois or to have class number 1) or of any finiteness or maximality hypotheses on the orders; such restrictions, if needed, must be stated before the theorem can be assessed.
minor comments (1)
  1. The abstract refers to 'a more general version of Furtwangler criterion' without citing the original statement or indicating precisely which variations are covered.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report and for highlighting points that require clarification. We respond to each major comment below and will revise the manuscript to address them.

read point-by-point responses

  1. Referee: The manuscript states the FTO and the multiplicative properties but supplies neither a definition of 'irreducible order' nor a proof of existence or uniqueness of the decomposition. Without these, the central claim cannot be verified.

    Authors: We acknowledge that the definition of an irreducible order and the full proof of existence and uniqueness were not presented with sufficient prominence or detail. An irreducible order is one that cannot be written as the intersection of two strictly larger orders. Existence and uniqueness follow from associating to each order its conductor ideal and applying the Chinese Remainder Theorem to factor the conductor into pairwise coprime parts, each corresponding to an irreducible order; the index multiplies accordingly. We will add an explicit definition and a self-contained proof section in the revised manuscript. revision: yes

  2. Referee: No explicit statement is given of the ambient number field (e.g., whether it is required to be Galois or to have class number 1) or of any finiteness or maximality hypotheses on the orders; such restrictions, if needed, must be stated before the theorem can be assessed.

    Authors: The result holds for an arbitrary number field K with no requirement that K be Galois or have class number one. The orders are any subrings of the ring of integers O_K that contain Z and have finite index in O_K; no further maximality assumptions are imposed. We will insert a preliminary paragraph stating these hypotheses explicitly before the statement of the FTO. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard background facts

full rationale

The abstract and context present the Fundamental Theorem of Orders as a new result on unique decomposition of orders into irreducible ones, with multiplicative index and coprime conductor properties. These rest on the standard facts that intersections of orders remain orders and that conductors/indices behave predictably under coprimeness—facts that hold for arbitrary number fields with no hidden restrictions or self-referential definitions. No equations, self-citations, fitted inputs presented as predictions, or ansatzes are visible that would reduce the claim to its own inputs by construction. The central claim therefore retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review based solely on abstract; the central claim introduces irreducible orders and the decomposition without supplying independent evidence or derivations.

axioms (1)
  • domain assumption Standard properties of orders as subrings of finite index in the ring of integers of a number field.
    The paper invokes the established theory of orders and conductors without re-deriving it.
invented entities (1)
  • irreducible orders no independent evidence
    purpose: To serve as the building blocks for the unique intersection decomposition of arbitrary orders.
    New concept introduced in the theorem; no independent evidence or prior reference is given in the abstract.

pith-pipeline@v0.9.1-grok · 5609 in / 1284 out tokens · 34246 ms · 2026-06-27T11:58:42.894747+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 12 canonical work pages

  1. [1]

    Gaurav Digambar Patil , Title =

  2. [2]

    , author=

    Unique Decomposition of Orders. , author=. , eprint=

  3. [3]

    , author=

    On the Ekedahl sieve for the singular locus of the discriminant polynomial. , author=. 2024 , eprint=

    2024

  4. [4]

    Squarefree Values Of Multivariable Polynomials , volume =

    Poonen, Bjorn , year =. Squarefree Values Of Multivariable Polynomials , volume =. Duke Mathematical Journal , doi =

  5. [5]

    Sitzungsberichte Akademie Wien , volume=

    Furtw. Sitzungsberichte Akademie Wien , volume=

  6. [6]

    Communications in Algebra , year=

    A Note on Conductor Ideals , author=. Communications in Algebra , year=

  7. [7]

    Birch, B. J. and Merriman, J. R. , title =. Proceedings of the London Mathematical Society , volume =. 1972 , month =. doi:10.1112/plms/s3-24.3.385 , url =

    work page doi:10.1112/plms/s3-24.3.385 1972

  8. [8]

    2014 , eprint=

    The geometric sieve and the density of squarefree values of invariant polynomials , author=. 2014 , eprint=

    2014

  9. [9]

    International Mathematics Research Notices , volume =

    Patil, Gaurav Digambar , title =. International Mathematics Research Notices , volume =. 2025 , month =. doi:10.1093/imrn/rnaf166 , url =

    work page doi:10.1093/imrn/rnaf166 2025

  10. [10]

    2024 , eprint=

    Weighted enumeration of number fields using Pseudo and Sudo maximal orders , author=. 2024 , eprint=

    2024

  11. [11]

    2010 , eprint=

    Parametrizing quartic algebras over an arbitrary base , author=. 2010 , eprint=

    2010

  12. [12]

    Bhargava, Manjul , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2004 , NUMBER =. doi:10.4007/annals.2004.159.217 , URL =

    work page doi:10.4007/annals.2004.159.217 2004

  13. [13]

    On the Distribution of Galois groups , journal =

    Gunter Malle , keywords =. On the Distribution of Galois groups , journal =. 2002 , issn =. doi:https://doi.org/10.1006/jnth.2001.2713 , url =

    work page doi:10.1006/jnth.2001.2713 2002

  14. [14]

    Experimental Mathematics , number =

    Gunter Malle , title =. Experimental Mathematics , number =

  15. [15]

    Acta Arithmetica , year=

    Decomposition of primes in non-maximal orders , author=. Acta Arithmetica , year=

  16. [16]

    Acta Arithmetica , year=

    On the number of equivalence classes of binary forms of given degree and given discriminant , author=. Acta Arithmetica , year=

  17. [17]

    Research in the Mathematical Sciences , year=

    Distribution of orders in number fields , author=. Research in the Mathematical Sciences , year=

  18. [18]

    Annals of Mathematics , year=

    The number of extensions of a number field with fixed degree and bounded discriminant , author=. Annals of Mathematics , year=

  19. [19]

    Bhargava, Manjul , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2004 , NUMBER =. doi:10.4007/annals.2004.159.1329 , URL =

    work page doi:10.4007/annals.2004.159.1329 2004

  20. [20]

    Mathematika , volume=

    The number of generators of the integers of a number field , author=. Mathematika , volume=. 1974 , publisher=. doi:10.1112/S0025579300008548 , url=

    work page doi:10.1112/s0025579300008548 1974

  21. [21]

    2021 , eprint=

    A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so , author=. 2021 , eprint=

    2021

  22. [22]

    2021 , eprint=

    A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so , author=. 2021 , eprint=

    2021

  23. [23]

    Bhargava, Manjul and Shankar, Arul and Wang, Xiaoheng , TITLE =. Invent. Math. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s00222-022-01098-w , URL =

    work page doi:10.1007/s00222-022-01098-w 2022

  24. [24]

    2009 , PAGES =

    Wood, Melanie Eggers Matchett , TITLE =. 2009 , PAGES =

    2009

  25. [25]

    Journal of the London Mathematical Society , volume=

    Rings and ideals parameterized by binary n -ic forms , author=. Journal of the London Mathematical Society , volume=. 2011 , publisher=. doi:10.1112/jlms/jdq074 , url=

    work page doi:10.1112/jlms/jdq074 2011

  26. [26]

    International Mathematics Research Notices , volume=

    Quartic Rings Associated to Binary Quartic Forms , author=. International Mathematics Research Notices , volume=. 2012 , publisher=. doi:10.1093/imrn/rnr070 , url=

    work page doi:10.1093/imrn/rnr070 2012

  27. [27]

    , author=

    Squarefree values of multivariable polynomials. , author=. Duke Mathematical Journal , volume=. 2003 , doi=

    2003

  28. [28]

    , author=

    Equality of Polynomial and Field Discriminants. , author=. Experimental Mathematics , volume=. 2007 , doi=

    2007

  29. [29]

    Forum of Mathematics, Sigma , volume=

    An improvement on Schmidt’s bound on the number of number fields of bounded discriminant and small degree , author=. Forum of Mathematics, Sigma , volume=. 2022 , publisher=. doi:10.1017/fms.2022.71 , url=

    work page doi:10.1017/fms.2022.71 2022

  30. [30]

    Binary forms and orders of algebraic number fields

    Nakagawa, Jin , journal =. Binary forms and orders of algebraic number fields. , url =

  31. [31]

    Shankar, Arul and Tsimerman, Jacob , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2023 , NUMBER =

    2023

  32. [32]

    and Heilbronn, H

    Davenport, H. and Heilbronn, H. , TITLE =. Proc. Roy. Soc. London Ser. A , FJOURNAL =. 1971 , NUMBER =. doi:10.1098/rspa.1971.0075 , URL =

    work page doi:10.1098/rspa.1971.0075 1971

  33. [33]

    Squarefree values of polynomial discriminants

    Manjul Bhargava and Arul Shankar and Xiaoheng Wang , year=. Squarefree values of polynomial discriminants. 2207.05592 , archivePrefix=

    arXiv

  34. [35]

    Mathematical Proceedings of the Cambridge Philosophical Society , author=

    Number fields without small generators , volume=. Mathematical Proceedings of the Cambridge Philosophical Society , author=. 2015 , pages=. doi:10.1017/S0305004115000298 , number=

    work page doi:10.1017/s0305004115000298 2015

  35. [36]

    The Stacks project , howpublished =

    The. The Stacks project , howpublished =