- Adjunction (field theory)
In

abstract algebra ,**adjunction**is a construction in field theory, where for a givenfield 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. For finite extensions :$A=\{a\_0,ldots,a\_n\}$ we often write :$F(a\_0,ldots,a\_n)$ instead of :$F(\{a\_0,ldots,a\_n\})$.simple extension **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".

* Thecomplex number s are constructed by adjunction of theimaginary unit to thereal number s, 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:$F(A)\; =\; igcup\_\{T\; in\; mathcal\{T\; F(T)$.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.

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