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arxiv: 2605.17315 · v1 · pith:TX3ZUXDWnew · submitted 2026-05-17 · 🧮 math.AC

Factorization in almost Dedekind domain

Gyu Whan Chang , Hyun Seung Choi This is my paper

Pith reviewed 2026-05-19 22:55 UTC · model grok-4.3

classification 🧮 math.AC
keywords almost Dedekind domainBézout domainfactorizationirreducible elementscyclotomic polynomialscountable productsprincipal primary ideals
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In the ring D formed as the union of F-adjoined p-power roots of X and their inverses, there are no irreducible elements when F is algebraically closed or finite of characteristic p, while countable prime factorizations exist for F equal to

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

The paper constructs D as the ascending union of rings D_n = F[X to the power 1 over p to the n, and X to the power minus 1 over p to the n]. This D is always a one-dimensional Bézout domain and becomes almost Dedekind exactly when the characteristic of F is not p. The authors track how polynomial irreducibility in each fixed D_n lifts or fails to lift to the full union, with the lifting condition expressed via divisibility by cyclotomic polynomials that satisfy explicit degree and order relations depending on the size of F. Special cases yield stronger conclusions: when F is algebraically closed or a finite field of characteristic p, D contains no irreducible elements at all. When F is the rationals and p equals 2, every nonzero nonunit factors as a countably infinite product of prime elements of D, and every proper nonzero principal ideal equals a unique countable intersection of principal primary ideals.

Core claim

D is a one-dimensional Bézout domain but not a Dedekind domain, and D is almost Dedekind if and only if char(F) ≠ p. An irreducible element of some D_n remains irreducible in D under explicit conditions on n and p. If F is algebraically closed or a finite field of characteristic p, then D has no irreducible elements. If F is a finite field of odd characteristic, an irreducible f(X) of D_0 is irreducible in D precisely when it divides a cyclotomic polynomial Φ_n(X) satisfying a stated relation involving |F| and deg(f). When F = Q and p = 2, every nonzero nonunit of D is a product of countably many prime elements and every proper nonzero principal ideal is a unique countable intersection of 1.

What carries the argument

The ring D defined as the direct union over n of the rings D_n = F[X^{1/p^n}, X^{-1/p^n}], which serves as the ambient almost Dedekind domain in which irreducibility criteria and countable factorizations are proved using cyclotomic polynomial factors and stabilization of elements at finite n.

If this is right

  • When char(F) ≠ p the ring D is almost Dedekind and therefore every nonzero prime ideal is maximal.
  • Irreducible elements of D_n persist as irreducible elements of D only for specific pairs (n, p) determined by the lifting criteria.
  • D admits no irreducible elements at all when F is algebraically closed or finite of characteristic p.
  • For finite F of odd characteristic, irreducibility in the union D reduces to divisibility of a cyclotomic polynomial satisfying an explicit numerical condition on |F| and degree.
  • When F = Q and p = 2 every nonzero nonunit factors into countably many primes and every proper principal ideal has a unique countable primary decomposition into principal primary ideals.

Where Pith is reading between the lines

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

  • The example shows that almost Dedekind domains can support infinite but still highly structured factorization that is absent from ordinary Dedekind domains.
  • The dependence on cyclotomic polynomials points to a possible link between irreducibility in D and Galois action on roots of unity in the base field.
  • Analogous unions over other root systems might produce further families of almost Dedekind domains whose factorization properties are completely determined by polynomial arithmetic at each finite stage.
  • It would be natural to test whether the countable-product property extends to other primes p or to other infinite base fields.

Load-bearing premise

The ring is built exactly as the ascending union of these Laurent polynomial rings over successive p-power roots of X, so that every element lies in some finite D_n and its behavior in the union can be read off from polynomial factorization in that D_n.

What would settle it

Exhibit a nonzero nonunit of D that cannot be written as a countable product of primes when F equals Q and p equals 2, or locate an irreducible element inside D when F is a finite field of characteristic p.

read the original abstract

Let $F$ be a field, $p$ a prime number, $X$ an indeterminate over $F$, $D_n =F[X^{\frac{1}{p^n}}, X^{-\frac{1}{p^n}}]$ for each integer $n \geq 0$ and $D = \bigcup\limits_{n\in\mathbb{N}_0}D_n.$ Then $D$ is a one-dimensional B{\'e}zout domain but not a Dedekind domain, and $D$ is an almost Dedekind domain if and only if char$(F) \neq p$. In this paper, we study the element-wise factorization properties of $D$. For example, we determine when an irreducible element of $D_n$ is an irreducible element of $D$, in terms of $n$ and $p$. In particular, we show that if $F$ is algebraically closed or a finite field of char$(F)=p$, then $D$ has no irreducible element. We also show that if $F$ is a finite field of odd characteristic, then an irreducible element $f(X)$ of $D_0$ is irreducible in $D$ if and only if it is a factor of a cyclotomic polynomial $\Phi_n(X)$ for some integer $n \geq 1$ which satisfies a certain equation in terms of $|F|$ and deg$(f(X))$. Finally, we introduce the notion of infinite product and we then show that if $F= \mathbb{Q}$ and $p=2$, every nonzero nonunit of $D$ can be written as a product of countably many prime elements of $D$ and every proper nonzero principal ideal of $D$ can be uniquely written as a countable intersection of principal primary ideals.

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

1 major / 2 minor

Summary. The manuscript defines D as the ascending union over n of the rings D_n = F[X^{1/p^n}, X^{-1/p^n}] for a field F and prime p. It proves that D is a one-dimensional Bézout domain which is almost Dedekind precisely when char(F) ≠ p. The paper then examines element factorization in D: it gives criteria (in terms of n and p) for when an irreducible of D_n remains irreducible in D; shows that D has no irreducible elements when F is algebraically closed or finite of characteristic p; provides a cyclotomic-polynomial criterion for irreducibility in D when F is finite of odd characteristic; and, for F = ℚ and p = 2, proves that every nonzero non-unit factors as a countable product of primes and that every proper nonzero principal ideal is a unique countable intersection of principal primary ideals.

Significance. The explicit ascending-union construction supplies concrete, verifiable examples of almost Dedekind domains whose factorization theory deviates markedly from the classical Dedekind case. The absence of irreducibles in the algebraically closed and characteristic-p cases, the cyclotomic criterion, and the countable-product and countable-intersection results illustrate how the failure of Noetherianness permits infinite factorizations while still retaining a form of uniqueness. The work therefore contributes concrete data to the study of factorization in Prüfer and almost Dedekind domains.

major comments (1)
  1. [§4] §4, Theorem 4.5: the cyclotomic criterion for irreducibility of elements of D_0 in the full ring D is stated in terms of an equation involving |F| and deg(f); the proof must explicitly confirm that the given degree condition prevents further splitting in all higher D_m for m > 0, rather than only in D_1.
minor comments (2)
  1. [Introduction] The introduction should list the main theorems with their numbers so that the abstract's claims can be located immediately.
  2. [§2] Notation: the monomial basis for elements of D_n is used repeatedly but never displayed explicitly; a short displayed formula would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive overall assessment. The single major comment is addressed point by point below. We agree that an explicit verification for all higher extensions will improve the clarity of the argument.

read point-by-point responses

  1. Referee: [§4] §4, Theorem 4.5: the cyclotomic criterion for irreducibility of elements of D_0 in the full ring D is stated in terms of an equation involving |F| and deg(f); the proof must explicitly confirm that the given degree condition prevents further splitting in all higher D_m for m > 0, rather than only in D_1.

    Authors: We thank the referee for this observation. The proof of Theorem 4.5 first establishes that the stated degree condition (derived from the order of F^* and the degree of the cyclotomic factor) implies that f remains irreducible in D_1. To make the argument complete for the full ring D, we will revise the proof by adding an explicit inductive step: suppose f factors nontrivially in some D_m with m > 1. Because each extension D_k / D_{k-1} is obtained by adjoining a p-th root and the minimal polynomial of any element of D_m over D_0 has degree a power of p, any factorization in D_m would descend to a factorization already in D_1, contradicting the irreducibility established there. The revised proof will therefore verify the condition directly for the entire tower rather than stopping at D_1. This clarification will be incorporated in the next version of the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivations rest on an explicit ring construction D = ∪ D_n with D_n = F[X^{1/p^n}, X^{-1/p^n}], combined with standard algebraic facts about field extensions, cyclotomic polynomials, and factorization in Bézout domains. Claims about the absence of irreducibles (when F is algebraically closed or finite of char p) follow directly from the ability to factor elements further in higher D_m levels via roots existing in F or perfectness. The cyclotomic irreducibility criterion and the countable product/intersection results for F=Q, p=2 are obtained from monomial descriptions and ideal intersections without any reduction to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The central results are self-contained against external benchmarks in commutative algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard axioms of fields and commutative rings, with the union construction of D as the primary defined object rather than an ad-hoc postulate.

axioms (2)
  • standard math F is a field, p a prime number, X an indeterminate over F
    Basic setup for defining the rings D_n and D.
  • domain assumption D equals the union over n of D_n where each D_n = F[X^{1/p^n}, X^{-1/p^n}]
    Central construction whose factorization properties are analyzed.

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