Fields of fractions for group algebras of free groups

Abstract

Let KF be the group algebra over the commutative field K of the free group F. It is proved that the field generated by KF in any Mal’cev-Neumann embedding for KF is the universal field of fractions U ( K F ) U(KF) of KF. Some consequences are noted. An example is constructed of an embedding K F ⊂ D KF \subset D into a field D with D ≄ U ( K F ) D\;\not \simeq \;U(KF) . It is also proved that the generalized free product of two free groups can be embedded in a field.