The question asks whether there exist non-isomorphic torsion-free abelian groups ( G ) and ( H ) such that their rational group algebra fields of fractions ( \mathbb{Q}(G) ) and ( \mathbb{Q}(H) ) are isomorphic.

Key observations:

1. For a free abelian group ( G ) of rank ( r ), the rational group algebra ( \mathbb{Q}[G] ) is isomorphic to the polynomial ring ( \mathbb{Q}[X_1, \ldo...

Short answer

• For free abelian groups G (finite or infinite rank): Q(G) determines the rank, hence if Q(G) ≅ Q(H) (as Q-fields), then G ≅ H.
• For rank-1 torsion-free abelian groups (i.e., subgroups of Q): again Q(G) determines G. In particular, if G and H are nonisomorphic rank-1 torsion-free abelian groups, then Q(G) ≇ Q(H).
• More generally, for finite rank torsion-free abelian groups, Q(G) at...

The present state of knowledge is the following.

1. Nobody knows a single example of two non-isomorphic torsion–free abelian groups whose rational group–algebra fields of fractions are isomorphic.
2. No proof is known that this can never happen.

In other words, the question is open.

Below is a summary of what is known and of the main ideas that have been tried.

A. Easy cases in which th...

Here is what is known, and why the answer to your question is “yes — in fact there are enormously many examples,” once you allow yourself to leave the world of purely transcendental extensions and pass to Hahn‐series (or more generally to fields which carry non‐trivial valuations).

1) The “purely transcendental” case is rigid.

– Gervasio Bastos–Viswanathan (Canad. Math. Bull. 31 (1988)) sh...

The Field of Fractions of Rational Group Algebras for Non-Isomorphic Torsion-Free Abelian Groups

Yes, there are non-isomorphic torsion-free abelian groups with isomorphic fields of fractions. I'll provide an explanation and examples.

An Explicit Example

Let me construct an example using the groups mentioned in the question:

If is a torsion-free abelian group such that $G \cong G \oplu...

An excellent and insightful question! The answer is remarkably definitive.

Are there non-isomorphic torsion free abelian groups and such that ?

No, there are no such examples. It is a theorem that for any two torsion-free abelian groups and , the fields of fractions and are isomorphic if and only if the groups ...

Yes, and in fact such examples are completely ubiquitous once one looks at torsion–free abelian groups of infinite rank.

Notation.
For a torsion–free abelian group (G) write
[ \Bbb Q[G]=\Bigl{\sum_{i=1}^{m}a_{i}g_{i};|;a_{i}\in\Bbb Q,;g_{i}\in G\Bigr},\qquad
\Bbb Q(G)=\operatorname{Frac}\Bbb Q[G]. ]

Step 1. A general fact about (\Bbb Q(G)).

Lemma 1. For every torsion–free ab...

<think> Okay, the user is asking about the field of fractions of the rational group algebra for torsion-free abelian groups. Specifically, they want to know if non-isomorphic torsion-free abelian groups can have isomorphic fraction fields, and if so, to see examples.

Hmm, this connects to that motivating question about groups isomorphic to their direct sum with Z² but not with Z. Interesting lin...

This is a very interesting question about the relationship between torsion-free abelian groups and their associated fields of fractions. Let me provide some insights and references.

Answer to the Main Question

Yes, there exist non-isomorphic torsion-free abelian groups and such that .

Key Results and Examples

1. The Fundamental Observation

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