mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

K

is pseudo algebraically closed if it satisfies certain properties which hold for

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

s. The concept was introduced by

James Ax James Burton Ax (10 January 1937 – 11 June 2006) was an American mathematician who made groundbreaking contributions in algebra and number theory using model theory. He shared, with Simon B. Kochen, the seventh Frank Nelson Cole Prize in ...

in 1967.Fried & Jarden (2008) p.218

Formulation

A field ''K'' is pseudo algebraically closed (usually abbreviated by PAC) if one of the following equivalent conditions holds: *Each

absolutely irreducible In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.. For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the integers ...

variety

V

defined over

K

has a

K

rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...

. *For each absolutely irreducible polynomial

f\in K_1,T_2,\cdots ,T_r,X /math> with \frac\not =0 and for each nonzero g\in K_1,T_2,\cdots ,T_r /math> there exists (\textbf,b)\in K^such that f(\textbf,b)=0 and g(\textbf)\not =0 .
*Each absolutely irreducible polynomial f\in K,X /math> has infinitely many K -rational points.
*If R is a finitely generated

integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...

over

K

with

quotient field In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...

which is regular over

K

, then there exist a

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

h:R\to K

such that

h(a) = a

for each

a \in K

Examples

* Algebraically closed fields and

separably closed In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynom ...

fields are always PAC. * Pseudo-finite fields and hyper-finite fields are PAC. * A non-principal

ultraproduct The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors ...

of distinct

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

s is (pseudo-finite and henceFried & Jarden (2008) p.449) PAC.Fried & Jarden (2008) p.192 Ax deduces this from the

Riemann hypothesis for curves over finite fields In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as :Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right) where is a non-singular -dimensional projective algeb ...

. * Infinite

algebraic extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...

s of finite fields are PAC.Fried & Jarden (2008) p.196 * ''The PAC Nullstellensatz''. The

absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' tha ...

G

of a field

K

is profinite, hence

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

, and hence equipped with a normalized

Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though ...

. Let

K

be a countable

Hilbertian field In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two fundam ...

and let

e

be a positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

. Then for almost all

e

-tuples

(\sigma_1,...,\sigma_e)\in G^e

, the fixed field of the

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

generated by the

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...

s is PAC. Here the phrase "almost all" means "all but a set of

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

zero".Fried & Jarden (2008) p.380 (This result is a consequence of Hilbert's irreducibility theorem.) * Let ''K'' be the maximal

totally real In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyno ...

Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...

of the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

s and ''i'' the square root of −1. Then ''K''(''i'') is PAC.

Properties

* The

Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...

of a PAC field is trivial,Fried & Jarden (2008) p.209 as any

Severi–Brauer variety In mathematics, a Severi–Brauer variety over a field ''K'' is an algebraic variety ''V'' which becomes isomorphic to a projective space over an algebraic closure of ''K''. The varieties are associated to central simple algebras in such a way ...

has a rational point. * The

of a PAC field is a

projective profinite group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...

; equivalently, it has

cohomological dimension In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory. Cohomological ...

at most 1.Fried & Jarden (2008) p.210 * A PAC field of characteristic zero is C1.Fried & Jarden (2008) p.462

References

* {{cite book , last1=Fried , first1=Michael D. , last2=Jarden , first2=Moshe , title=Field arithmetic , edition=3rd revised , series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge , volume=11 , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, year=2008 , isbn=978-3-540-77269-9 , zbl=1145.12001 Algebraic geometry Field (mathematics)