P-adically closed field

Last updated December 30, 2022

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.^[1]

Definition

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

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

(Note that the value group of K may be larger than that of F since it may contain infinitely large elements over the latter.)

The formally p-adic fields can be viewed as an analogue of the formally real fields.

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

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

If F is p-adically closed, then:^[2]

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

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

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

The Kochen operator

If K is a field equipped with a valuation v satisfying the hypothesis and with the notations introduced in the previous paragraph, define the Kochen operator by:

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

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

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

First-order theory

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

Notes

↑ Ax & Kochen (1965)
↑ Jarden & Roquette (1980), lemma 4.1

Related Research Articles

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

In mathematics, particularly in algebra, a field extension is a pair of fields $such that the operations of E are those of F restricted to E . In this case, F is an extension field of E and E is a subfield of F . For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.$

In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

In mathematics, the $p$ -adic number system for any prime number $p$ extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two $p$ -adic numbers are considered to be close when their difference is divisible by a high power of $p$ : the higher the power, the closer they are. This property enables $p$ -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

In mathematics, a field K is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C are also defined to be local fields; this is the convention we will adopt below. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields.

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

In algebra, a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.

In algebraic geometry, motives is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.

In mathematics, the ring of integers of an algebraic number field $is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of, the ring is always a subring of .$

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x⁻¹ belongs to D.

<span class="mw-page-title-main">Puiseux series</span> Power series with rational exponents

In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series

In mathematics, a field F is called quasi-algebraically closed (or C₁) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper (Tsen 1936); and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper (Lang 1952). The idea itself is attributed to Lang's advisor Emil Artin.

In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by Dennis Sullivan (1977) and Daniel Quillen (1969). This simplification of homotopy theory makes certain calculations much easier.

In mathematics, Hahn series are a type of formal infinite series. They are a generalization of Puiseux series and were first introduced by Hans Hahn in 1907. They allow for arbitrary exponents of the indeterminate so long as the set supporting them forms a well-ordered subset of the value group. Hahn series were first introduced, as groups, in the course of the proof of the Hahn embedding theorem and then studied by him in relation to Hilbert's second problem.

In mathematics, an algebraic number field is an extension field $of the field of rational numbers such that the field extension has finite degree . Thus is a field that contains and has finite dimension when considered as a vector space over .$

This is a glossary of algebraic geometry.

In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in the sense of generalised function.

References

Ax, James; Kochen, Simon (1965). "Diophantine problems over local fields. II. A complete set of axioms for 𝑝-adic number theory". Amer. J. Math. The Johns Hopkins University Press. 87 (3): 631–648. doi:10.2307/2373066. JSTOR 2373066.
Kochen, Simon (1969). "Integer valued rational functions over the 𝑝-adic numbers: A 𝑝-adic analogue of the theory of real fields". Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967). American Mathematical Society. pp. 57–73.
Kuhlmann, F.-V. (2001) [1994], "p-adically closed field", Encyclopedia of Mathematics , EMS Press , retrieved 2009-02-03
Jarden, Moshe; Roquette, Peter (1980). "The Nullstellensatz over 𝔭-adically closed fields". J. Math. Soc. Jpn. 32 (3): 425–460. doi: 10.2969/jmsj/03230425 .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Ax & Kochen (1965)

[2] Jarden & Roquette (1980), lemma 4.1

[1]

[2]