Editing P-adic number (section)

== Algebraic closure ==

<math>\Q_p</math> contains <math>\Q</math> and is a field of [[characteristic (algebra)|characteristic]] {{math|0}}.

{{anchor|not_orderable}}Because {{math|0}} can be written as sum of squares,<ref>According to [[Hensel's lemma#Examples|Hensel's lemma]] <math>\Q_2</math> contains a square root of {{math|−7}}, so that <math>2^2 +1^2+1^2+1^2+\left(\sqrt{-7}\right)^2 = 0 ,</math> and if {{math|''p'' > 2}} then also by Hensel's lemma <math>\Q_p</math> contains a square root of {{math|1 − ''p''}}, thus 
<math>(p-1)\times 1^2 +\left(\sqrt{1-p}\right)^2 = 0 .</math></ref> <math>\Q_p</math>  cannot be turned into an [[Ordered field#Orderability of fields|ordered field]].

The field of [[real numbers]] <math>\R</math> has only a single proper [[algebraic extension]]: the [[complex numbers]] <math>\C</math>.  In other words, this [[quadratic extension]] is already [[algebraically closed field|algebraically closed]]. By contrast, the [[algebraic closure]] of <math>\Q_p</math>, denoted <math>\overline{\Q_p},</math> has infinite degree,<ref>{{Harv|Gouvêa|1997|loc=Corollary 5.3.10}}</ref> that is, <math>\Q_p</math> has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the {{mvar|p}}-adic valuation to <math>\overline{\Q_p},</math> the latter is not (metrically) complete.<ref>{{Harv|Gouvêa|1997|loc=Theorem 5.7.4}}</ref><ref name=C149>{{Harv|Cassels|1986|p=149}}</ref> Its (metric) completion is called <math>\C_p</math> or <math>\Omega_p</math>.<ref name=C149/><ref name=K13>{{Harv|Koblitz|1980|p=13}}</ref> Here an end is reached, as <math>\C_p</math> is algebraically closed.<ref name=C149/><ref>{{Harv|Gouvêa|1997|loc=Proposition 5.7.8}}</ref> However unlike <math>\C</math> this field is not [[locally compact]].<ref name=K13/>

<math>\C_p</math> and <math>\C</math> are isomorphic as rings,<ref>Two algebraically closed fields are isomorphic if and only if they have the same characteristic and transcendence degree (see, for example Lang’s ''Algebra'' X §1), and both <math>\C_p</math> and <math>\C</math> have characteristic zero and the cardinality of the continuum.</ref> so we may regard <math>\C_p</math> as <math>\C</math> endowed with an exotic metric. The proof of existence of such a field isomorphism relies on the [[axiom of choice]], and does not provide an explicit example of such an isomorphism (that is, it is not [[constructive proof|constructive]]).

If <math>K</math> is any finite [[Galois extension]] of <math>\Q_p,</math> the [[Galois group]] <math>\operatorname{Gal} \left(K/\Q_p \right)</math> is [[solvable group|solvable]]. Thus, the Galois group <math>\operatorname{Gal} \left(\overline{\Q_p}/ \Q_p \right)</math> is [[prosolvable]].