As another application, we show that there are (many) proper subfields of the complex number field C which are (as fields) isomorphic to C. For the proof, take a transcendence basis "S" of C/Q. "S" is an infinite (even uncountable) set, so there exist (many) maps "f": "S" → "S" which are injective but not surjective. Any such map can be extended to a field homomorphism Q("S") → Q("S") which is not surjective. Such a field homomorphism can in turn be extended to the algebraic closure C, and the resulting field homomorphisms C → C are not surjective.