In mathematics, a pseudo-finite field ''F'' is an infinite model of the
first-order theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
of
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. 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 of an algebraic set is an algebraic subset that is irred ...
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 fact ...
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 in ...
, year=2008 , isbn=978-3-540-77269-9 , zbl=1145.12001 , pages=448–453
Model theory
Field (mathematics)