The Inversion Height of the Free Field is Infinite

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 k[H] into the Malcev-Neumann series ring is also infinite. This answer in the affirmative a question posed by Neumann in 1949 [27, p. 215]. 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 commutative 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.