Quantum geometry of field extensions

We introduce a new kind of topological gauge configuration or "soliton'' as sociated to the extension C superset of or equal to R of the real numbers t o the complex ones. These configurations describe zero-curvature gauge fiel ds and nontrivial cohomology over the real line, but with a quantum choice of differential calculus. In general, the quantum differential 1-forms on t he line with coordinate algebra k[x] are in correspondence with field exten sions of k, and the quantum cohomology detects the nontriviality of the ext ension. (C) 1999 American Institute of Physics. [S0022-2488(99)04002-5].