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