Definitions

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

$A=\left\{a_0,ldots,a_n\right\}$
we often write
$F\left(a_0,ldots,a_n\right)$
$F\left(\left\{a_0,ldots,a_n\right\}\right)$.

## 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.

## Properties

Given a field extension E/F and a subset A of E. Let $mathcal\left\{T\right\}$ be the family of all finite subsets of A then

$F\left(A\right) = bigcup_\left\{T in mathcal\left\{T\right\}\right\} F\left(T\right)$.
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(MN) = 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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