A characterization of monomorphisms of skew fields (Q1587846)

The author proves that if \(\mathcal{F}\) is a skew field then a nonzero function \(f: \mathcal{F} \to \mathcal{F}\) satisfying the equation \[ f((x+y)(x-y)^{-1})(f(x) - f(y)) = f(x) + f(y),\quad x \not= y, \] is necessarily a homeomorphism of \(\mathcal{F}\) if, and only if, char \(\mathcal{F}\) \( \not= 2 \) and card \(\mathcal{F}\) \(\not= 5\). The cases where card \(\mathcal{F}\) = \(p\) (a prime number) or \(\mathcal{F}\) is a special extension of \(\mathcal{Q}\) were considered by \textit{K. S. Sarkaria} [ibid. 60, No. 1-2, 137-141 (2000; Zbl 0966.39012)].