- Field of definition
In

mathematics , the**field of definition**of analgebraic variety "V" is essentially the smallest field to which the coefficients of thepolynomial s 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 subscript of "alg", e.g. the algebraic closure of "k" is "k"^{alg}. The symbols**Q**,**R**,**C**, and**F**_{"p"}represent, respectively, the field ofrational numbers , the field ofreal numbers , the field ofcomplex numbers , and thefinite 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}) withprojective space of dimension "n" − 1 over "k"^{alg}, and by insisting that all polynomials be homogeneous.A

**"k"-**is the zero-locus inalgebraic set **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 thelinear 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 islinearly independent over "k" is also linearly independent over "k"^{alg}. In other words those extensions of "k" arelinearly 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.**Examples**# The zero-locus of "x"

_{1}^{2}+ "x"_{2}^{2}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}+ i"x"_{2}and "x"_{1}- i"x"_{2}.

#With**F**_{"p"}("t") atranscendental extension of**F**_{"p"}, the polynomial "x"_{1}^{"p"}- "t" equals ("x"_{1}- "t"^{1/"p"})^{"p"}in the polynomial ring (**F**_{"p"}("t"))^{alg}["x"_{1}] . The**F**_{"p"}("t")-algebraic set "V" defined by "x"_{1}^{"p"}- "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}- "t"^{1/"p"}.

# Thecomplex projective line is a projective**R**-variety. (In fact, it is a variety with**Q**as its minimal field of definition.) Viewing thereal projective line as being the equator on the Riemann sphere, the coordinate-wise action ofcomplex 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"_{1}^{2}+ "x"_{2}^{2}+ "x"_{3}^{2}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") → (2"ab", "a"^{2}-"b"^{2}, -i("a"^{2}+"b"^{2})). Identifying "W" with the Riemann sphere using this map, the coordinate-wise action ofcomplex 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.**cheme-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 "V" ×_{Spec("k")}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 thefinal 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}+ i"x"_{2}- (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"_{1}^{2}+ "x"_{2}^{2}+ 2"x"_{3}^{2}- 2"x"_{1}"x"_{3}- 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}+ i"x"_{2}- (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}, …, "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" ×_{Spec("k")}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" ×_{Spec("k")}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" ×_{Spec("k")}Spec("k"^{alg}) on "U" ×_{Spec("k")}Spec("k"^{alg}) whose residues are constant on each Gal("k"^{alg}/"k")-orbit in "U" ×_{Spec("k")}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}, …, "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"}- "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}- "t"^{1/"p"}, by "x"_{1}^{"p"}- "t", or, indeed, by "x"_{1}- "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.

coauthors = Moshe Jarden

title = Field Arithmetic

publisher =Springer

date = 2005

pages = 780

doi = 10.1007/b138352

isbn = 354022811X

** 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 = 0817630651

** 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 = 354063293X

** 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.

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