section_path: Characterizations, generalizations and related notions/Algebraic characterization
section_name: Algebraic characterization
paragraph_text: Algebraic characterization
  
  The field C {\displaystyle \mathbb {C} } has the following three properties:
  
  It can be shown that any field having these properties is isomorphic (as a field) to C. {\displaystyle \mathbb {C}.} For example, the algebraic closure of the field Q p {\displaystyle \mathbb {Q} _{p}} of the p -adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields). Also, C {\displaystyle \mathbb {C} } is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that C {\displaystyle \mathbb {C} } contains many proper subfields that are isomorphic to C {\displaystyle \mathbb {C} }.