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Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$? In the case of $n=1$, there are only trivial units: $K[F_1]^\...

Let $F = \langle x_1, \dots, x_n \rangle$ be the free group on $n$ generators and $R = \mathbb Z$. The Fox derivatives $\frac{\partial}{\partial x_i} \colon F \to R[F]$ are the unique derivations ...