# Adjunction (field theory)

In abstract algebra, **adjunction** is a construction in field theory, where for a given field extension *E*/*F*, subextensions between *E* and *F* are constructed.

## Definition

Let *E* be a field extension of a field *F*. Given a set of elements *A* in the larger field *E* we denote by *F*(*A*) the smallest subextension which contains the elements of *A*. We say *F*(*A*) is constructed by **adjunction** of the elements *A* to *F* or **generated** by *A*.

If *A* is finite we say *F*(*A*) is **finitely generated** and if *A* consists of a single element we say *F*(*A*) is a **simple extension**. The primitive element theorem states a finite separable extension is simple.

A finitely generated extension may not be a finite extension. For example, **Q**(*e*) is a finitely generated extension of the rationals, but of infinite degree since *e* is transcendental. In a sense, a finitely generated extension is a transcendental generalization of a finite extension since, if the generators in *A* are all algebraic, then *F*(*A*) is a finite extension of *F*. Because of this, most examples come from algebraic geometry.

A subextension of a finitely generated field extension is also a finitely generated extension.^{[1]}

## Notes

*F*(*A*) consists of all those elements of *E* that can be constructed using a finite number of field operations +, -, *, / applied to elements from *F* and *A*. For this reason *F*(*A*) is sometimes called the **field of rational expressions** in *F* and *A*.

## Examples

- Given a field extension
*E*/*F*then*F*(Ø) =*F*and*F*(*E*) =*E*. - The complex numbers are constructed by adjunction of the imaginary unit to the real numbers, that is
**C**=**R**(i).

## Properties

Given a field extension *E*/*F* and a subset *A* of *E*, let be the family of all finite subsets of *A*. Then

- .

In other words the adjunction of any set can be reduced to a union of adjunctions of finite sets.

Given a field extension *E*/*F* and two subsets *N*, *M* of *E* then *K*(*M* ∪ *N*) = (*K*(*M*))(*N*) = (*K*(*N*))(*M*). This shows that any adjunction of a finite set can be reduced to a successive adjunction of single elements.

## References

- ↑ Kolchin, E. R. (1973),
*Differential Algebra & Algebraic Groups*, Pure and Applied Mathematics,**54**, Academic Press, p. 112, ISBN 9780080873695.