A characterization of the complex number field (Q1590074)

Building on previous work, the author proves the following theorem. Let \(N\) be a nearring with additive group isomorphic to \(\mathbb{R}^2\), the Euclidean plane. Assume that the multiplication on \(N\) is continuous, but not necessarily associative. If \(N\) contains a central element \(c\) such that \(-c^2\) is a left identity, then \(N\) is isomorphic to the field of complex numbers. It is proved by example that the hypothesis ``\(c\) central'' cannot be dispensed with.