# 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 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.

## 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$_1$ + ix$_2$ and x$_1$ - ix$_2$.
* 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) → (2ab, 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 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 ×$_{Spec(k)}$ 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$_1$ + ix$_2$ - (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$+ 2x$_3$$^2$- 2x$_1$x$_3$ - 2x$_2$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$_1$ + ix$_2$ - (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.