The concept of the "field of definition" is a fundamental idea in mathematics, particularly in areas such as algebraic geometry, number theory, and field theory. It refers to the smallest field over which a given mathematical object can be defined or described. This concept is crucial for understanding the structure and properties of algebraic varieties, polynomials, and other mathematical entities.

1. Introduction to the Field of Definition

In mathematics, many objects are defined over a field, which is a set equipped with addition, subtraction, multiplication, and division operations (except division by zero). Examples of fields include the rational numbers (ℚ), the real numbers (ℝ), and the complex numbers (ℂ). The field of definition is the smallest field that contains all the coefficients or parameters necessary to define a particular object.

For example, consider a polynomial equation like ( f(x) = x^2 + \sqrt{2}x + 1 ). The coefficients of this polynomial are ( 1 ), ( \sqrt{2} ), and ( 1 ). The smallest field containing these coefficients is ( \mathbb{Q}(\sqrt{2}) ), which is the field obtained by adjoining ( \sqrt{2} ) to the rational numbers. Therefore, the field of definition for this polynomial is ( \mathbb{Q}(\sqrt{2}) ).

2. Field of Definition in Algebraic Geometry

In algebraic geometry, the field of definition is particularly important when studying algebraic varieties, which are geometric manifestations of solutions to polynomial equations. An algebraic variety is said to be defined over a field ( k ) if its defining equations have coefficients in ( k ).

For instance, consider the elliptic curve defined by the equation ( y^2 = x^3 + ax + b ), where ( a ) and ( b ) are elements of a field ( k ). The field of definition for this elliptic curve is ( k ), because the coefficients ( a ) and ( b ) are in ( k ).

However, sometimes an algebraic variety can be defined over a smaller field than the one initially used. For example, the curve ( y^2 = x^3 + 1 ) can be defined over the rational numbers ( \mathbb{Q} ), even though it might initially be considered over the real numbers ( \mathbb{R} ). The field of definition is thus the smallest field over which the variety can be described.

3. Field of Definition in Number Theory

In number theory, the field of definition is often used in the context of algebraic numbers and Galois theory. An algebraic number is a root of a non-zero polynomial equation with rational coefficients. The field of definition for an algebraic number is the smallest field containing all the coefficients of its minimal polynomial over ( \mathbb{Q} ).

For example, the number ( \sqrt{2} ) is a root of the polynomial ( x^2 - 2 ), which has rational coefficients. The field of definition for ( \sqrt{2} ) is ( \mathbb{Q} ), because the coefficients of its minimal polynomial are in ( \mathbb{Q} ).

In Galois theory, the field of definition is closely related to the concept of a splitting field. The splitting field of a polynomial is the smallest field extension over which the polynomial factors into linear factors. The field of definition for the polynomial is thus contained within its splitting field.

4. Field of Definition in Field Theory

In field theory, the field of definition is used to describe the smallest field over which a given field extension is defined. For example, consider a field extension ( L/K ), where ( L ) is a larger field containing ( K ). The field of definition for this extension is ( K ), because ( L ) is constructed by adjoining elements to ( K ).

However, if ( L ) can be generated over a smaller subfield ( k ) of ( K ), then ( k ) is the field of definition for the extension ( L/K ). This concept is important when studying the structure of field extensions and their automorphisms.

5. Field of Definition in Representation Theory

In representation theory, the field of definition is used to describe the smallest field over which a representation of a group can be defined. A representation of a group ( G ) over a field ( k ) is a homomorphism from ( G ) to the general linear group ( \text{GL}(V) ), where ( V ) is a vector space over ( k ).

The field of definition for a representation is the smallest field ( k ) such that all the matrix entries of the representation are in ( k ). For example, if a representation of a group can be realized with matrices whose entries are rational numbers, then the field of definition for that representation is ( \mathbb{Q} ).

6. Field of Definition in Algebraic Number Theory

In algebraic number theory, the field of definition is used to describe the smallest field over which an algebraic number field can be defined. An algebraic number field is a finite extension of the rational numbers ( \mathbb{Q} ). The field of definition for an algebraic number field is ( \mathbb{Q} ), because it is generated by adjoining algebraic numbers to ( \mathbb{Q} ).

However, sometimes an algebraic number field can be defined over a smaller field. For example, the field ( \mathbb{Q}(\sqrt{2}) ) is defined over ( \mathbb{Q} ), but it can also be considered as a field extension of ( \mathbb{Q}(\sqrt{2}) ) itself. In this case, the field of definition is still ( \mathbb{Q} ), because ( \mathbb{Q}(\sqrt{2}) ) is generated by adjoining ( \sqrt{2} ) to ( \mathbb{Q} ).

7. Field of Definition in Algebraic Geometry and Scheme Theory

In scheme theory, a generalization of algebraic geometry, the field of definition is used to describe the smallest field over which a scheme can be defined. A scheme is a mathematical object that generalizes the notion of an algebraic variety. The field of definition for a scheme is the smallest field over which the scheme's structure sheaf and morphisms can be defined.

For example, consider a scheme ( X ) defined over a field ( k ). The field of definition for ( X ) is ( k ), because all the defining equations and morphisms of ( X ) are defined over ( k ). However, if ( X ) can be defined over a smaller subfield ( k' ) of ( k ), then ( k' ) is the field of definition for ( X ).

8. Field of Definition in Arithmetic Geometry

In arithmetic geometry, the field of definition is used to study the properties of algebraic varieties over number fields. A number field is a finite extension of the rational numbers ( \mathbb{Q} ). The field of definition for an algebraic variety over a number field is the smallest number field over which the variety can be defined.

For example, consider an elliptic curve defined over a number field ( K ). The field of definition for this elliptic curve is ( K ), because the coefficients of its defining equation are in ( K ). However, if the elliptic curve can be defined over a smaller number field ( k ) contained in ( K ), then ( k ) is the field of definition for the elliptic curve.

9. Field of Definition in Galois Cohomology

In Galois cohomology, the field of definition is used to study the cohomology groups associated with Galois extensions of fields. The field of definition for a Galois extension ( L/K ) is ( K ), because the Galois group ( \text{Gal}(L/K) ) acts on ( L ) as a ( K )-vector space.

However, if the Galois extension ( L/K ) can be defined over a smaller field ( k ) contained in ( K ), then ( k ) is the field of definition for the extension. This concept is important when studying the structure of Galois cohomology groups and their applications in number theory.

10. Conclusion

The field of definition is a fundamental concept in many areas of mathematics, including algebraic geometry, number theory, field theory, and representation theory. It refers to the smallest field over which a given mathematical object can be defined or described. Understanding the field of definition is crucial for studying the structure and properties of algebraic varieties, polynomials, field extensions, and representations of groups.

In summary, the field of definition provides a way to understand the minimal algebraic context in which a mathematical object exists. It allows mathematicians to study objects over the smallest possible field, which often simplifies the analysis and reveals deeper properties of the object. Whether in algebraic geometry, number theory, or representation theory, the field of definition is a key tool for understanding the underlying structure of mathematical objects.