Pith Number

pith:TX3ZUXDW

pith:2026:TX3ZUXDWGYNU4IUAXPLHGXMSL7

not attested not anchored not stored refs resolved

Factorization in almost Dedekind domain

Gyu Whan Chang, Hyun Seung Choi

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

arxiv:2605.17315 v1 · 2026-05-17 · math.AC

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\usepackage{pith}
\pithnumber{TX3ZUXDWGYNU4IUAXPLHGXMSL7}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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Claims

C1strongest claim

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=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.

C2weakest assumption

The ring D is almost Dedekind precisely when char(F) ≠ p, and the irreducibility criteria rely on properties of cyclotomic polynomials and field extensions in the specific construction of the D_n.

C3one line summary

In the almost Dedekind domain D built from field F and prime p, the paper gives conditions for irreducibles in D_n to stay irreducible in D, shows D has no irreducibles for algebraically closed F or finite F of characteristic p, and for F=Q and p=2 proves every nonzero nonunit factors into countably

References

46 extracted · 46 resolved · 0 Pith anchors

[1] D. D. Anderson, GCD Domains, Gauss’ Lemma, and Contents of Polynomials , Non- Noetherian Commutative Ring Theory, Mathematics and Its Applications 520 (Kluwer Aca- demic Publishers, Dordrecht, 2000) p 2000

[2] D. D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility , Proc. Amer. Math. Soc. 109(4) (1990), 907-913 1990

[3] D. D. Anderson and M. Zafrullah, A generalization of unique factorization , Bollettino U.M.I. 9-A (1995), 401-413 1995

[4] Arnold, Krull dimension in power series rings , Trans 1973

[5] , Power series rings over Pr¨ ufer domains, Pacific J. Math. 44 (1973), 1-11 1973

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:51.726169Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

9df79a5c76361b4e2280bbd6735d925fefb064eac36be012511f05778e97fbb9

Aliases

arxiv: 2605.17315 · arxiv_version: 2605.17315v1 · doi: 10.48550/arxiv.2605.17315 · pith_short_12: TX3ZUXDWGYNU · pith_short_16: TX3ZUXDWGYNU4IUA · pith_short_8: TX3ZUXDW

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/TX3ZUXDWGYNU4IUAXPLHGXMSL7 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 9df79a5c76361b4e2280bbd6735d925fefb064eac36be012511f05778e97fbb9

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8f67b635e447b2d5899465d6ea83a4ce7625ab152a855c3484080ea6fe1b0fc7",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AC",
    "submitted_at": "2026-05-17T08:13:30Z",
    "title_canon_sha256": "3531ebc24a3757fb6af382e2d5bc506a0d775154b69a8a6022b2d3fde7ad1ecd"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17315",
    "kind": "arxiv",
    "version": 1
  }
}