bibkey: aliabadi2025matchabilityii
authors: Mohsen Aliabadi
year: 2025
title: Conditions for matchability in groups and field extensions II
doi: 10.7151/dmgaa.1461
url: https://www.dmgaa.uz.zgora.pl/publish/pdf.php?doi=1461
claim: "Conjecture 38 proposes an uncapped lower bound for all subspace products under a premise concerning proper finite intermediate fields."
strata_touched:
  - D5/S3/Factorization/Galois/AliabadiSubspaceProductRefutation
license: citation-only
triage: anchor

Subspace-product lower bound

Discussiones Mathematicae General Algebra and Applications 45 (2025), 135–157, section 5.1, Conjecture 38, printed page 154 / PDF page 20. The same statement is Conjecture 5.1 in arXiv:2309.14664v1, section 5.1, PDF page 14, and versions v2 and v3, PDF page 15.

Let K ⊂ L be a field extension and A,B ⊂ L be a K-subspace of L of finite positive dimensions. Assume further that for every a,b ∈ L and every nontrivial proper finite intermediate subfield K ⊂ M ⊊ L, we have dim_K(aM ∩ A) + dim_K(bM ∩ B) ≤ [M:K] + 1. Then for all K-subspaces S ⊂ A and T ⊂ B, the following holds dim_K⟨ST⟩ ≥ dim_K S + dim_K T − 1.

Section 2.2 (arXiv PDF page 7; journal printed page 143 / PDF page 9) defines ⟨ST⟩ as the K-linear span of the actual product set in L. A linear translate aM consists of the products am with m in M. Nontrivial means M properly contains K; proper means M differs from L. There is no restriction excluding finite ambient extensions or imposing a bound on the sum of the two subspace dimensions. Section 5.2 concerns a separate conjecture on Chowla subspaces.

The equal-dimension and 1-not-in-B conditions in the introduction define linear matchability; Conjecture 38 does not invoke matchability as a hypothesis. It states finite positive dimensions for A and B, without requiring equality or that either space contain 1. The counterexample nevertheless has A=B, S=T, all four spaces exclude 1, and every indicated subspace containment is strict. The scalar quantifiers include zero. The word nontrivial excludes the base field, while proper excludes L.

In the quotient Q[X]/(X^7−2), Eisenstein’s criterion gives a field of degree seven. Prime degree excludes every intermediate field strictly between Q and the quotient. Take A=B=span(alpha,…,alpha^6) and S=T=span(alpha,…,alpha^5), where alpha is the class of X. The dimensions are six and five respectively, and these are proper subspaces with 1 absent. The intermediate-field premise is therefore vacuous, whereas any product span has dimension at most seven. The proposed lower bound would require dimension at least nine. Exact equality of the product span with the whole field is unnecessary for this contradiction.

The formal statement retains all field extensions, both finite positive dimensions, every scalar pair, every proper nontrivial finite intermediate field, and all pairs of contained subspaces. The selected proper subspaces also satisfy the strict reading of the source’s ordinary subset symbols.

Verified locator

• DOI: https://doi.org/10.7151/dmgaa.1461
• Journal: https://www.dmgaa.uz.zgora.pl/publish/pdf.php?doi=1461 ; section 5.1, Conjecture 38, printed page 154 / PDF page 20.
• https://arxiv.org/pdf/2309.14664v1 ; section 5.1, Conjecture 5.1, PDF page 14.
• https://arxiv.org/pdf/2309.14664v2 ; section 5.1, Conjecture 5.1, PDF page 15.
• https://arxiv.org/pdf/2309.14664v3 ; section 5.1, Conjecture 5.1, PDF page 15.

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