P-adically closed field

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Let ${\displaystyle K}$ be the field ${\displaystyle \mathbb {Q} }$ of rational numbers and ${\displaystyle v}$ be its usual ${\displaystyle p}$-adic valuation (with ${\displaystyle v(p)=1}$). If ${\displaystyle F}$ is a (not necessarily algebraic) extension field of ${\displaystyle K}$, itself equipped with a valuation ${\displaystyle w}$, we say that ${\displaystyle (F,w)}$ is formally p-adic when the following conditions are satisfied:

• ${\displaystyle w}$ extends ${\displaystyle v}$ (that is, ${\displaystyle w(x)=v(x)}$ for all ${\displaystyle x\in K}$),
• the residue field of ${\displaystyle w}$ coincides with the residue field of ${\displaystyle v}$ (the residue field being the quotient of the valuation ring ${\displaystyle \{x\in F:w(x)\geq 0\}}$ by its maximal ideal ${\displaystyle \{x\in F:w(x)>0\}}$),
• the smallest positive value of ${\displaystyle w}$ coincides with the smallest positive value of ${\displaystyle v}$ (namely 1, since ${\displaystyle v}$ was assumed to be normalized): in other words, a uniformizer for ${\displaystyle K}$ remains a uniformizer for ${\displaystyle F}$.

Note that the value group of ${\displaystyle K}$ may be larger than that of ${\displaystyle F}$ since it may contain infinitely large elements over the latter.

Thus the formally ${\displaystyle p}$-adic fields can be viewed as an analogue of the formally real fields.

For example, the field ${\displaystyle \mathbb {Q} (i)}$ of Gaussian rationals, if equipped with the valuation ${\displaystyle w}$ given by ${\displaystyle w(2+i)=1}$ (and ${\displaystyle w(2-i)=0}$) is formally 5-adic (the place ${\displaystyle v=5}$ of the rationals splits in two places of the Gaussian rationals since ${\displaystyle x^{2}+1}$ factors over the residue field with 5 elements, and ${\displaystyle w}$ 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 ${\displaystyle w}$) 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 ${\displaystyle w}$ on it which extends the 3-adic valuation is given by ${\displaystyle w(3)=1}$ and its residue field has 9 elements.

When ${\displaystyle F}$ is formally ${\displaystyle p}$-adic but that there does not exist any proper algebraic formally ${\displaystyle p}$-adic extension of ${\displaystyle F}$, then ${\displaystyle F}$ is said to be p-adically closed. For example, the field of ${\displaystyle p}$-adic numbers is ${\displaystyle p}$-adically closed, and so is the algebraic closure of the rationals inside it (the field of ${\displaystyle p}$-adic algebraic numbers).

If ${\displaystyle F}$ is ${\displaystyle p}$-adically closed, then^[2]

• there is a unique valuation ${\displaystyle w}$ on ${\displaystyle F}$ which makes ${\displaystyle F}$ ${\displaystyle p}$-adically closed (so it is legitimate to say that ${\displaystyle F}$, rather than the pair ${\displaystyle (F,w)}$, is ${\displaystyle p}$-adically closed),
• ${\displaystyle F}$ is Henselian with respect to this place (that is, its valuation ring is so),
• the valuation ring of ${\displaystyle F}$ is exactly the image of the Kochen operator (see below),
• the value group of ${\displaystyle F}$ is an extension by ${\displaystyle \mathbb {Z} }$ (the value group of ${\displaystyle K}$) of a divisible group, with the lexicographical order.

The first statement is an analogue of the fact that the order of a real-closed field is uniquely determined by the algebraic structure.

The definitions given above can be copied to a more general context: if ${\displaystyle K}$ is a field equipped with a valuation ${\displaystyle v}$ such that

• the residue field of ${\displaystyle K}$ is finite (call ${\displaystyle q}$ its cardinality and ${\displaystyle p}$ its characteristic),
• the value group of ${\displaystyle v}$ admits a smallest positive element (call it 1, and say ${\displaystyle \pi }$ is a uniformizer, i.e. ${\displaystyle v(\pi )=1}$),
• ${\displaystyle K}$ has finite absolute ramification, i.e., ${\displaystyle v(p)}$ is finite (that is, a finite multiple of ${\displaystyle v(\pi )=1}$),

then we can speak of formally ${\displaystyle v}$-adic fields (or ${\displaystyle {\mathfrak {p}}}$-adic if ${\displaystyle {\mathfrak {p}}}$ is the ideal corresponding to ${\displaystyle v}$) and ${\displaystyle v}$-adically complete fields. These hypotheses are notably satisfied for the field of rationals, with the prime number ${\displaystyle q=\pi =p}$ having valuation 1.

If ${\displaystyle K}$ is a field equipped with a valuation ${\displaystyle v}$ satisfying the hypothesis and with the notations introduced in the previous paragraph, define the Kochen operator by

${\displaystyle \gamma (z)={\frac {1}{\pi }}\,{\frac {z^{q}-z}{(z^{q}-z)^{2}-1}}}$

(when ${\displaystyle z^{q}-z\neq \pm 1}$). It is easy to check that ${\displaystyle \gamma (z)}$ always has non-negative valuation. The Kochen operator can be thought of as a ${\displaystyle p}$-adic (or ${\displaystyle v}$-adic) analogue of the square function in the real case.

An extension field ${\displaystyle F}$ of ${\displaystyle K}$ is formally ${\displaystyle v}$-adic if and only if ${\displaystyle 1/\pi }$ does not belong to the subring generated over the value ring of ${\displaystyle K}$ by the image of the Kochen operator on ${\displaystyle F}$. This is an analogue of the statement that a field is formally real when ${\displaystyle -1}$ is not a sum of squares.

The first-order theory of ${\displaystyle p}$-adically closed fields (here we are restricting ourselves to the ${\displaystyle p}$-adic case, i.e., ${\displaystyle K}$ is the field of rationals and ${\displaystyle v}$ is the ${\displaystyle p}$-adic valuation) is complete and model complete, and if we slightly enrich the language it admits quantifier elimination. Thus, one can define ${\displaystyle p}$-adically closed fields as those whose first-order theory is elementarily equivalent to that of ${\displaystyle \mathbb {Q} _{p}}$.