# Field of definition

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 insisting that all polynomials 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

**is a**

*k*-variety*k*-algebraic set that is irreducible, i.e. is not the union of two strictly smaller

*k*-algebraic sets. A

**is a regular function between**

*k*-morphism*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.

## 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*) a transcendental 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}. - 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*_{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 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

**is an irreducible**

*k*-variety*k*-algebraic set. A

**is a morphism between**

*k*-morphism*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* ×_{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 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} + 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

- Fried, Michael D.; Moshe Jarden (2005).
*Field Arithmetic*. Springer. p. 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.

- Kunz, Ernst (1985).
*Introduction to Commutative Algebra and Algebraic Geometry*. Birkhäuser. p. 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.

- Mumford, David (1999).
*The Red Book of Varieties and Schemes*. Springer. pp. 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.