In mathematics, a pseudo-finite field ''F'' is an infinite model of the

first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...

theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...

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. This is equivalent to the condition that ''F'' is quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely

irreducible variety In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (fo ...

over ''F'' has a point defined over ''F''). Every hyperfinite field is pseudo-finite and every pseudo-finite field is quasifinite. Every 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 factor ...

of finite fields is pseudo-finite. Pseudo-finite fields were introduced by .

References

* * {{citation , 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 ...

, year=2008 , isbn=978-3-540-77269-9 , zbl=1145.12001 , pages=448–453 Model theory Field (mathematics)