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. For finite extensions :$A=\{a\_0,ldots,a\_n\}$ we often write :$F(a\_0,ldots,a\_n)$ instead of :$F(\{a\_0,ldots,a\_n\})$.

Notes

"F"("A") consists of all those elements of "F" 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 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 $mathcal\{T\}$ 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 subset "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.