Resumen de The inversion height of the free field is infinite

Dolors Herbera i Espinal , Javier Sánchez

• Let XX be a finite set with at least two elements, and let kk be any commutative field. We prove that the inversion height of the embedding k⟨X⟩↪Dk⟨X⟩↪D , where DD denotes the universal (skew) field of fractions of the free algebra k⟨X⟩k⟨X⟩ , is infinite. Therefore, if HH denotes the free group on XX , the inversion height of the embedding of the group algebra kHkH into the Malcev–Neumann series ring is also infinite. This answers in the affirmative a question posed by Neumann (Trans Am Math Soc 66:202–252, 1949). We also give an infinite family of examples of non-isomorphic fields of fractions of k⟨X⟩k⟨X⟩ with infinite inversion height. We show that the universal field of fractions of a crossed product of a field by the universal enveloping algebra of a free Lie algebra is a field of fractions constructed by Cohn (and later by Lichtman). This extends a result by A. Lichtman.