Let $k$ be a field of characteristic $p$ and $G$ be a torsion free abelian group . Then the group ring $k[G]$ is an integral domain , let $k(G)$ denote its field of fractions . Then can we say anything about the transcendence degree of $k(G)$ over $F_p$ in terms of $k$ and/or $G$ ? What about the same question for field $k$ of characteristic $0$ ? An answer to this might help in solving https://math.stackexchange.com/questions/2338911/fraction-field-of-group-ring-of-field-over-torsion-free-abelian-group

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