Reconstruction of Torsion-Free Abelian Groups from Rational Group Fields

Technical Summary: Reconstruction of Torsion-Free Abelian Groups from Rational Group Fields

Problem Statement
The paper addresses a question posed by Rickard regarding the isomorphism of rational group fields associated with torsion-free abelian groups. Specifically, it investigates whether the isomorphism of the fraction fields of rational group algebras, $K_{\mathbb{Q}}(G) \cong K_{\mathbb{Q}}(H)$, implies the isomorphism of the underlying groups $G \cong H$. This problem is particularly significant in the context of infinite-rank groups, where standard invariants may fail to distinguish structures. Previous work by Bastos and Viswanathan established that $K_{\mathbb{Q}}(G)$ is purely transcendental over $\mathbb{Q}$ if and only if $G$ is free abelian, but this did not resolve whether arbitrary group fields reflect group isomorphism.

Methodology
The proof strategy relies on a structural analysis of the multiplicative groups of the fraction fields, specifically focusing on the "monomial defect group." The methodology proceeds through the following logical stages:

1. Definition of the Defect Group: The authors define the monomial defect group as the quotient $\Delta_k(G) = K_k(G)^\times / (k^\times X^G)$, where $X^G$ represents the subgroup of formal monomials. This group measures the deviation of the field's multiplicative structure from the group algebra's monomial structure.
2. Freeness of the Defect Group: The core structural input is the proof that for any field $k$ of characteristic zero and any torsion-free abelian group $G$, the defect group $\Delta_k(G)$ is a free abelian group.
   • One-Variable Case: The proof begins with the Puiseux field $F(t^{\mathbb{Q}})$. By analyzing the transition from $F(t^{1/n!})$ to $F(t^{1/(n+1)!})$, the authors demonstrate that the substitution of irreducible polynomials yields split inclusions because the substituted irreducibles are square-free in characteristic zero. This establishes that the defect group for the one-variable Puiseux quotient is free abelian.
   • General Case: Using a transfinite decomposition of the divisible hull $V = \mathbb{Q} \otimes_{\mathbb{Z}} G$, the authors extend the freeness result to arbitrary torsion-free abelian groups. They utilize the fact that subgroups of free abelian groups are free abelian and that the defect group of a subgroup injects into the defect group of the overgroup.
3. Reconstruction via Intersection: Given a field isomorphism $\phi: K_{\mathbb{Q}}(G) \to K_{\mathbb{Q}}(H)$, the authors compare the images of the monomial subgroups $M \le G$ and $N \le H$ defined by the preservation of monomials under $\phi$.
   • The kernels of the induced defect maps are identified as $M$ and $N$.
   • The quotients $G/M$ and $H/N$ are shown to embed into the free defect groups $\Delta_{\mathbb{Q}}(H)$ and $\Delta_{\mathbb{Q}}(G)$, respectively. Consequently, these quotients are free abelian.
   • The restriction of $\phi$ identifies the subfields $K_{\mathbb{Q}}(M)$ and $K_{\mathbb{Q}}(N)$.
   • By comparing the relative transcendence degrees of the extensions $K_{\mathbb{Q}}(G)/K_{\mathbb{Q}}(M)$ and $K_{\mathbb{Q}}(H)/K_{\mathbb{Q}}(N)$, the authors prove that the free quotients $G/M$ and $H/N$ have the same rank.
   • Since the extensions split (due to the projectivity of free abelian groups), $G \cong M \oplus (G/M)$ and $H \cong N \oplus (H/N)$. The isomorphism of the components $M \cong N$ and the equality of the ranks of the free parts yield $G \cong H$.

Key Contributions and Results

• The Reconstruction Theorem (Theorem 1.1): The primary result establishes that for all torsion-free abelian groups $G$ and $H$, $K_{\mathbb{Q}}(G) \cong K_{\mathbb{Q}}(H)$ if and only if $G \cong H$. This provides a negative answer to Rickard's question, confirming that the rational group field is a complete invariant for torsion-free abelian groups.
• Structural Insight: The paper provides a self-contained proof that the monomial defect group $\Delta_k(G)$ is free abelian in characteristic zero. This result isolates the specific role of characteristic zero (specifically the square-free nature of derivatives in the Puiseux construction) and contrasts it with the positive characteristic case, where the defect group fails to be free (e.g., containing a subgroup isomorphic to $\mathbb{Z}[1/p]$).
• Stabilization Profile: The authors derive a corollary regarding "exact stabilization." They prove that rational-function stabilization of group fields ($K_{\mathbb{Q}}(G)(t_i) \cong K_{\mathbb{Q}}(H)(u_j)$) exactly records the free stabilization of the groups themselves ($G \oplus \mathbb{Z}^{|I|} \cong H \oplus \mathbb{Z}^{|J|}$).
• Rickard's Example: Applying the theorem to Rickard's bounded-sequence group $A$ (where $A \cong A \oplus \mathbb{Z}^2$ but $A \not\cong A \oplus \mathbb{Z}$), the paper constructs a field $F = K_{\mathbb{Q}}(A)$ such that $F \cong F(x, y)$ but $F \not\cong F(x)$. This demonstrates that field isomorphism does not imply isomorphism after adding a single variable, even though it does after adding two.

Significance and Scope
The paper claims significance in providing a definitive reconstruction of torsion-free abelian groups from their rational group fields without imposing hypotheses on rank, countability, unique factorization, or cancellation. The proof is noted for being "self-contained" and for explicitly identifying where the characteristic zero assumption is utilized.

The authors explicitly limit the scope of their results:

• The reconstruction applies strictly to torsion-free abelian groups; groups with torsion generally yield group algebras with zero divisors, rendering the standard fraction field undefined.
• The result relies on characteristic zero. The paper notes that the central freeness theorem fails in positive characteristic, leaving the reconstruction problem for finite fields open.
• The proof establishes the existence of an isomorphism between the abstract groups but does not claim that the standard monomial subgroup is definable without parameters within the field, nor does it construct a canonical copy of $G$ inside $K_{\mathbb{Q}}(G)$. The reconstruction relies on comparing two monomial frames given a pre-existing field isomorphism.

The work acknowledges the historical context provided by May's earlier results on multiplicative group freeness and Rickard's formulation of the problem, while noting that the discovery and structural organization of the proofs were assisted by AI tools, with all mathematical arguments independently verified by the authors.