In mathematics, a p-adically closed field is a field that enjoys a closure property that is a close analogue for p-adic fields to what real closure is to the real field. They were introduced by James Ax and Simon B. Kochen in 1965.
Definition Let K {\displaystyle K} be the field Q {\displaystyle \mathbb {Q} } of rational numbers and v {\displaystyle v} be its usual p {\displaystyle p} -adic valuation (with v ( p ) = 1 {\displaystyle v(p)=1} ). If F {\displaystyle F} is a (not necessarily algebraic) extension field of K {\displaystyle K} , itself equipped with a valuation w {\displaystyle w} , we say that ( F , w ) {\displaystyle (F,w)} is formally p-adic when the following conditions are satisfied:
w {\displaystyle w} extends v {\displaystyle v} (that is, w ( x ) = v ( x ) {\displaystyle w(x)=v(x)} for all x ∈ K {\displaystyle x\in K} ), the residue field of w {\displaystyle w} coincides with the residue field of v {\displaystyle v} (the residue field being the quotient of the valuation ring { x ∈ F : w ( x ) ≥ 0 } {\displaystyle \{x\in F:w(x)\geq 0\}} by its maximal ideal { x ∈ F : w ( x ) > 0 } {\displaystyle \{x\in F:w(x)>0\}} ), the smallest positive value of w {\displaystyle w} coincides with the smallest positive value of v {\displaystyle v} (namely 1, since v {\displaystyle v} was assumed to be normalized): in other words, a uniformizer for K {\displaystyle K} remains a uniformizer for F {\displaystyle F} . Note that the value group of K {\displaystyle K} may be larger than that of F {\displaystyle F} since it may contain infinitely large elements over the latter. Thus the formally p {\displaystyle p} -adic fields can be viewed as an analogue of the formally real fields. For example, the field Q ( i ) {\displaystyle \mathbb {Q} (i)} of Gaussian rationals, if equipped with the valuation w {\displaystyle w} given by w ( 2 + i ) = 1 {\displaystyle w(2+i)=1} (and w ( 2 − i ) = 0 {\displaystyle w(2-i)=0} ) is formally 5-adic (the place v = 5 {\displaystyle v=5} of the rationals splits in two places of the Gaussian rationals since x 2 + 1 {\displaystyle x^{2}+1} factors over the residue field with 5 elements, and w {\displaystyle w} is one of these places). The field of 5-adic numbers (which contains both the rationals and the Gaussian rationals embedded as per the place w {\displaystyle w} ) is also formally 5-adic. On the other hand, the field of Gaussian rationals is not formally 3-adic for any valuation, because the only valuation w {\displaystyle w} on it which extends the 3-adic valuation is given by w ( 3 ) = 1 {\displaystyle w(3)=1} and its residue field has 9 elements. When F {\displaystyle F} is formally p {\displaystyle p} -adic but that there does not exist any proper algebraic formally p {\displaystyle p} -adic extension of F {\displaystyle F} , then F {\displaystyle F} is said to be p-adically closed. For example, the field of p {\displaystyle p} -adic numbers is p {\displaystyle p} -adically closed, and so is the algebraic closure of the rationals inside it (the field of p {\displaystyle p} -adic algebraic numbers). If F {\displaystyle F} is p {\displaystyle p} -adically closed, then
there is a unique valuation w {\displaystyle w} on F {\displaystyle F} which makes F {\displaystyle F} p {\displaystyle p} -adically closed (so it is legitimate to say that F {\displaystyle F} , rather than the pair ( F , w ) {\displaystyle (F,w)} , is p {\displaystyle p} -adically closed),
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