mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 . In other words, a field is algebraically closed if the fundamental theorem of algebra ...

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 integ ...

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 fiel ...

. *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, 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 setting for studying divisibilit ...

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 fiel ...

which is

regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

over

K

, then there exist a

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

h:R\to K

such that

h(a) = a

for each

a \in K

Examples

* Algebraically closed fields and

separably closed In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1 ...

fields are always PAC. *

Pseudo-finite field In mathematics, a pseudo-finite field ''F'' is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that ''F'' is quasi-finite (perfect with a unique extension of every positive degree) and pseudo a ...

s 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 fact ...

of distinct

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

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 mathematics, 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^)^k\right) where is a non-singular -dimensional projective algebr ...

. * 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, every element of is a root of a non-zero polynomial with coefficients in . A field extens ...

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'' ...

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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

, 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 (mathematics), measure was introduced by Alfr� ...

. 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 fundame ...

and let

e

be a positive

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

. 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, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

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 automorphism ...

s is PAC. Here the phrase "almost all" means "all but a set of measure 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 (mathematics), image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one zero ...

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 (for example, The set of all ...

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

Properties

* The

Brauer group In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...

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 (mathematics), 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 algebr ...

has a rational point. * The

of a PAC field is a projective profinite group; 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)