Editing Complex number (section)

===Algebraic characterization===
The field <math>\Complex</math> has the following three properties:
* First, it has [[characteristic (algebra)|characteristic]] 0. This means that {{math|1=1 + 1 + ⋯ + 1 ≠ 0}} for any number of summands (all of which equal one).
* Second, its [[transcendence degree]] over <math>\Q</math>, the [[prime field]] of <math>\Complex,</math> is the [[cardinality of the continuum]].
* Third, it is [[algebraically closed]] (see above).
It can be shown that any field having these properties is [[isomorphic]] (as a field) to <math>\Complex.</math> For example, the [[algebraic closure]] of the field <math>\Q_p</math> of the [[p-adic number|{{mvar|p}}-adic number]] also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).<ref>{{cite book
 | last = Marker | first = David
 | editor1-last = Marker | editor1-first = D.
 | editor2-last = Messmer | editor2-first = M.
 | editor3-last = Pillay | editor3-first = A.
 | contribution = Introduction to the Model Theory of Fields
 | contribution-url = https://projecteuclid.org/euclid.lnl/1235423155
 | isbn = 978-3-540-60741-0
 | mr = 1477154
 | pages = 1–37
 | publisher = Springer-Verlag | location = Berlin
 | series = Lecture Notes in Logic
 | title = Model theory of fields
 | volume = 5
 | year = 1996}}</ref> Also, <math>\Complex</math> 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 <math>\Complex</math> contains many proper subfields that are isomorphic to <math>\Complex</math>.