Field of definition

{{More footnotes|date=November 2021}}

In mathematics, the '''field of definition''' of an algebraic variety ''V'' is essentially the smallest field to which the coefficients of the polynomials defining ''V'' can belong. Given polynomials, with coefficients in a field ''K'', it may not be obvious whether there is a smaller field ''k'', and other polynomials defined over ''k'', which still define ''V''.

The issue of field of definition is of concern in diophantine geometry.

==Notation==

Throughout this article, ''k'' denotes a field. The algebraic closure of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of ''k'' is ''k''^alg. The symbols '''Q''', '''R''', '''C''', and '''F'''_''p'' represent, respectively, the field of rational numbers, the field of real numbers, the field of complex numbers, and the finite field containing ''p'' elements. Affine ''n''-space over a field ''F'' is denoted by '''A'''^''n''(''F'').

==Definitions for affine and projective varieties==

Results and definitions stated below, for affine varieties, can be translated to projective varieties, by replacing '''A'''^''n''(''k''^alg) with projective space of dimension ''n'' − 1 over ''k''^alg, and by requiring all polynomials to be homogeneous.

A '''''k''-algebraic set''' is the zero-locus in '''A'''^''n''(''k''^alg) of a subset of the polynomial ring ''k''[''x''₁, ..., ''x''_''n'']. A '''''k''-variety''' is a ''k''-algebraic set that is irreducible, i.e. is not the union of two strictly smaller ''k''-algebraic sets. A '''''k''-morphism''' is a regular function between ''k''-algebraic sets whose defining polynomials' coefficients belong to ''k''.

One reason for considering the zero-locus in '''A'''^''n''(''k''^alg) and not '''A'''^''n''(''k'') is that, for two distinct ''k''-algebraic sets ''X''₁ and ''X''₂, the intersections ''X''₁ ∩ '''A'''^''n''(''k'') and ''X''₂ ∩ '''A'''^''n''(''k'') can be identical; in fact, the zero-locus in '''A'''^''n''(''k'') of any subset of ''k''[''x''₁, ..., ''x''_''n''] is the zero-locus of a ''single'' element of ''k''[''x''₁, ..., ''x''_''n''] if ''k'' is not algebraically closed.

A ''k''-variety is called a '''variety''' if it is ''absolutely irreducible'', i.e. is not the union of two strictly smaller ''k''^alg-algebraic sets. A variety ''V'' is '''defined over ''k''''' if every polynomial in ''k''^alg[''x''₁, ..., ''x''_''n''] that vanishes on ''V'' is the linear combination (over ''k''^alg) of polynomials in ''k''[''x''₁, ..., ''x''_''n''] that vanish on ''V''. A ''k''-algebraic set is also an ''L''-algebraic set for infinitely many subfields ''L'' of ''k''^alg. A '''field of definition''' of a variety ''V'' is a subfield ''L'' of ''k''^alg such that ''V'' is an ''L''-variety defined over ''L''.

Equivalently, a ''k''-variety ''V'' is a variety defined over ''k'' if and only if the function field ''k''(''V'') of ''V'' is a regular extension of ''k'', in the sense of Weil. That means every subset of ''k''(''V'') that is linearly independent over ''k'' is also linearly independent over ''k''^alg. In other words those extensions of ''k'' are linearly disjoint.

André Weil proved that the intersection of all fields of definition of a variety ''V'' is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition.{{Citation needed|reason=Non-trivial theorem|date=February 2026}} However, this does not apply to abstract varieties, and in fact, counterexamples exist.<ref>{{cite web |url=https://mathoverflow.net/questions/19478/fields-of-definition-of-a-variety |title=Fields of definition of a variety |work=Mathoverflow |date=2010-03-27 |access-date=2026-02-10 }}</ref>

==Examples==

# The zero-locus of ''x''₁²+&nbsp;''x''₂² is both a '''Q'''-variety and a '''Q'''^alg-algebraic set but neither a variety nor a '''Q'''^alg-variety, since it is the union of the '''Q'''^alg-varieties defined by the polynomials ''x''₁&nbsp;+&nbsp;i''x''₂ and ''x''₁&nbsp;−&nbsp;i''x''₂. # <div id="non-reduced-example"></div>With '''F'''_''p''(''t'') a transcendental extension of '''F'''_''p'', the polynomial ''x''₁^''p''&nbsp;−&nbsp;''t'' equals (''x''₁&nbsp;−&nbsp;''t''^1/''p'')&nbsp;^''p'' in the polynomial ring ('''F'''_''p''(''t''))^alg[''x''₁]. The '''F'''_''p''(''t'')-algebraic set ''V'' defined by ''x''₁^''p''&nbsp;−&nbsp;''t'' is a variety; it is absolutely irreducible because it consists of a single point. But ''V'' is not defined over '''F'''_''p''(''t''), since ''V'' is also the zero-locus of ''x''₁&nbsp;−&nbsp;''t''^1/''p''. # The complex projective line is a projective '''R'''-variety. (In fact, it is a variety with '''Q''' as its minimal field of definition.) Viewing the real projective line as being the equator on the Riemann sphere, the coordinate-wise action of complex conjugation on the complex projective line swaps points with the same longitude but opposite latitudes. # The projective '''R'''-variety ''W'' defined by the homogeneous polynomial ''x''₁²&nbsp;+&nbsp;''x''₂²&nbsp;+&nbsp;''x''₃² is also a variety with minimal field of definition '''Q'''. The following map defines a '''C'''-isomorphism from the complex projective line to ''W'': (''a'',''b'')&nbsp;→&nbsp;(2''ab'',&nbsp;''a''²−''b''²,&nbsp;−i(''a''²+''b''²)). Identifying ''W'' with the Riemann sphere using this map, the coordinate-wise action of complex conjugation on ''W'' interchanges opposite points of the sphere. The complex projective line cannot be '''R'''-isomorphic to ''W'' because the former has ''real points'', points fixed by complex conjugation, while the latter does not.

==Scheme-theoretic definitions==

One advantage of defining varieties over arbitrary fields through the theory of schemes is that such definitions are intrinsic and free of embeddings into ambient affine ''n''-space.

A '''''k''-algebraic set''' is a separated and reduced scheme of finite type over Spec(''k''). A '''''k''-variety''' is an irreducible ''k''-algebraic set. A '''''k''-morphism''' is a morphism between ''k''-algebraic sets regarded as schemes over Spec(''k'').

To every algebraic extension ''L'' of ''k'', the ''L''-algebraic set associated to a given ''k''-algebraic set ''V'' is the fiber product of schemes ''V''&nbsp;×_Spec(''k'')&nbsp;Spec(''L''). A ''k''-variety is absolutely irreducible if the associated ''k''^alg-algebraic set is an irreducible scheme; in this case, the ''k''-variety is called a '''variety'''. An absolutely irreducible ''k''-variety is '''defined over ''k''''' if the associated ''k''^alg-algebraic set is a reduced scheme. A '''field of definition''' of a variety ''V'' is a subfield ''L'' of ''k''^alg such that there exists a ''k''∩''L''-variety ''W'' such that ''W'' ×_{Spec(''k''∩''L'')} Spec(''k'') is isomorphic to ''V'' and the final object in the category of reduced schemes over ''W'' ×_{Spec(''k''∩''L'')} Spec(''L'') is an ''L''-variety defined over ''L''.

Analogously to the definitions for affine and projective varieties, a ''k''-variety is a variety defined over ''k'' if the stalk of the structure sheaf at the generic point is a regular extension of ''k''; furthermore, every variety has a minimal field of definition.

One disadvantage of the scheme-theoretic definition is that a scheme over ''k'' cannot have an ''L''-valued point if ''L'' is not an extension of ''k''. For example, the rational point (1,1,1) is a solution to the equation ''x''₁&nbsp;+&nbsp;i''x''₂&nbsp;-&nbsp;(1+i)''x''₃ but the corresponding '''Q'''[i]-variety ''V'' has no Spec('''Q''')-valued point. The two definitions of ''field of definition'' are also discrepant, e.g. the (scheme-theoretic) minimal field of definition of ''V'' is '''Q''', while in the first definition it would have been '''Q'''[i]. The reason for this discrepancy is that the scheme-theoretic definitions only keep track of the polynomial set ''up to change of basis''. In this example, one way to avoid these problems is to use the '''Q'''-variety Spec('''Q'''[''x''₁,''x''₂,''x''₃]/(''x''₁²+&nbsp;''x''₂²+&nbsp;2''x''₃²-&nbsp;2''x''₁''x''₃&nbsp;-&nbsp;2''x''₂''x''₃)), whose associated '''Q'''[i]-algebraic set is the union of the '''Q'''[i]-variety Spec('''Q'''[i][''x''₁,''x''₂,''x''₃]/(''x''₁&nbsp;+&nbsp;i''x''₂&nbsp;-&nbsp;(1+i)''x''₃)) and its complex conjugate.

==Action of the absolute Galois group==

The absolute Galois group Gal(''k''^alg/''k'') of ''k'' naturally acts on the zero-locus in '''A'''ⁿ(''k''^alg) of a subset of the polynomial ring ''k''[''x''₁,&nbsp;...,&nbsp;''x''_''n'']. In general, if ''V'' is a scheme over ''k'' (e.g. a ''k''-algebraic set), Gal(''k''^alg/''k'') naturally acts on ''V''&nbsp;×_Spec(''k'')&nbsp;Spec(''k''^alg) via its action on Spec(''k''^alg).

When ''V'' is a variety defined over a perfect field ''k'', the scheme ''V'' can be recovered from the scheme ''V''&nbsp;×_Spec(''k'')&nbsp;Spec(''k''^alg) together with the action of Gal(''k''^alg/''k'') on the latter scheme: the sections of the structure sheaf of ''V'' on an open subset ''U'' are exactly the sections of the structure sheaf of ''V''&nbsp;×_Spec(''k'')&nbsp;Spec(''k''^alg) on ''U''&nbsp;×_Spec(''k'')&nbsp;Spec(''k''^alg) whose residues are constant on each Gal(''k''^alg/''k'')-orbit in ''U''&nbsp;×_Spec(''k'')&nbsp;Spec(''k''^alg). In the affine case, this means the action of the absolute Galois group on the zero-locus is sufficient to recover the subset of ''k''[''x''₁,&nbsp;...,&nbsp;''x''_''n''] consisting of vanishing polynomials.

In general, this information is not sufficient to recover ''V''. In the example of the zero-locus of ''x''₁^''p''&nbsp;−&nbsp;''t'' in ('''F'''_''p''(''t''))^alg, the variety consists of a single point and so the action of the absolute Galois group cannot distinguish whether the ideal of vanishing polynomials was generated by ''x''₁&nbsp;−&nbsp;''t''^1/''p'', by ''x''₁^''p''&nbsp;−&nbsp;''t'', or, indeed, by ''x''₁&nbsp;−&nbsp;''t''^1/''p'' raised to some other power of ''p''.

For any subfield ''L'' of ''k''^alg and any ''L''-variety ''V'', an automorphism σ of ''k''^alg will map ''V'' isomorphically onto a σ(''L'')-variety.

==Further reading==

* {{cite book | last = Fried | first = Michael D. |author-link= Michael D. Fried |author2=Moshe Jarden | title = Field Arithmetic | publisher = Springer | date = 2005 | pages = 780 | doi = 10.1007/b138352 | isbn = 3-540-22811-X }} ** The terminology in this article matches the terminology in the text of Fried and Jarden, who adopt Weil's nomenclature for varieties. The second edition reference here also contains a subsection providing a dictionary between this nomenclature and the more modern one of schemes. * {{cite book | last = Kunz | first = Ernst | title = Introduction to Commutative Algebra and Algebraic Geometry | publisher = Birkhäuser | date = 1985 | pages = 256 | isbn = 0-8176-3065-1 }} ** Kunz deals strictly with affine and projective varieties and schemes but to some extent covers the relationship between Weil's definitions for varieties and Grothendieck's definitions for schemes. * {{cite book | last = Mumford | first = David | authorlink = David Mumford | title = The Red Book of Varieties and Schemes | publisher = Springer | date = 1999 | pages = 198–203 | doi = 10.1007/b62130 | isbn = 3-540-63293-X }} ** Mumford only spends one section of the book on arithmetic concerns like the field of definition, but in it covers in full generality many scheme-theoretic results stated in this article.

==References== {{Reflist}}

Category:Diophantine geometry Category:Algebraic geometry