Adjunction (field theory)

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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.

1 Definition
2 Notes
3 Examples
4 Properties

[edit] 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

$F(\{a_0,\ldots,a_n\})$ .

[edit] 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.

[edit] 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).

[edit] 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

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

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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