The inversion height of the free field is infinite

Abstract

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 $$k\langle X\rangle \hookrightarrow D$$ , where $$D$$ denotes the universal (skew) field of fractions of the free algebra $$k\langle X\rangle $$ , 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 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 $$k\langle X\rangle $$ 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.