The inversion height of the free field is infinite

© 2014, Springer Basel. Let X be a finite set with at least two elements, and let k be any commutative field. We prove that the inversion height of the embedding K ⟨X⟩ → D, where D denotes the universal (skew) field of fractions of the free algebra K ⟨X⟩, is infinite. Therefore, if H denotes the free group on X, the inversion height of the embedding of the group algebra kH 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⟩ 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.