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Algebraic closure

Q p {\displaystyle \mathbb {Q} _{p}} contains Q {\displaystyle \mathbb {Q} } and is a field of characteristic 0.

Because 0 can be written as sum of squares, Q p {\displaystyle \mathbb {Q} _{p}} cannot be turned into an ordered field.

The field of real numbers R {\displaystyle \mathbb {R} } has only a single proper algebraic extension: the complex numbers C {\displaystyle \mathbb {C} }. In other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of Q p {\displaystyle \mathbb {Q} _{p}}, denoted Q p ¯, {\displaystyle {\overline {\mathbb {Q} _{p}}},} has infinite degree, that is, Q p {\displaystyle \mathbb {Q} _{p}} has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the p -adic valuation to Q p ¯, {\displaystyle {\overline {\mathbb {Q} _{p}}},} the latter is not (metrically) complete. Its (metric) completion is called C p {\displaystyle \mathbb {C} _{p}} or Ω p {\displaystyle \Omega _{p}}. Here an end is reached, as C p {\displaystyle \mathbb {C} _{p}} is algebraically closed. However unlike C {\displaystyle \mathbb {C} } this field is not locally compact.

C p {\displaystyle \mathbb {C} _{p}} and C {\displaystyle \mathbb {C} } are isomorphic as rings, so we may regard C p {\displaystyle \mathbb {C} _{p}} as C {\displaystyle \mathbb {C} } 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).

If K {\displaystyle K} is any finite Galois extension of Q p, {\displaystyle \mathbb {Q} _{p},}, the Galois group Gal (K / Q p) {\displaystyle \operatorname {Gal} \left(K/\mathbb {Q} _{p}\right)} is solvable. Thus, the Galois group Gal (Q p ¯ / Q p) {\displaystyle \operatorname {Gal} \left({\overline {\mathbb {Q} _{p}}}/\mathbb {Q} _{p}\right)} is prosolvable.