nLab field extension


This page is about adjunctions of a set to a field in field theory. For the notion of adjunction in 2-category theory, see adjunction.



If a subset kk of a field KK is a subfield, then we call the larger field KK an extension of the smaller field kk.

More generally, if kKk \to K is any ring homomorphism between fields, then it must be an injection, so we may treat it as a field extension.

Adjunctions of a set to a field

Let kk be a field, let KK be a field extension of kk, and let SKS \subseteq K be a subset of KK. kKk \subseteq K is also a subset of KK. Then the adjunction of SS to kk, or the field generated by SS over kk, is the initial subfield k(S)Kk(S) \subseteq K such that kKk \subseteq K and SKS \subseteq K.


Factorization system

Every field extension can be factorized as a purely transcendental extension followed by an algebraic extension. Indeed, by Zorn's lemma, we may construct a transcendence basis (i.e. maximal algebraically independent set) BB, and the purely transcendental part is the subfield generated by BB.

Unfortunately, this does not yield an orthogonal factorization system: given a field KK, we may form the field K(x)K (x) of rational functions over KK, which is a purely transcendental extension of KK, and we may form the algebraic closure K(x)¯\overline{K (x)}, which is an algebraic extension of K(x)K (x); but we have the following commutative diagram,

K K(x 2) K(x) K(x)¯\array{ K & \to & K (x^2) \\ \downarrow & & \downarrow \\ K (x) & \to & \overline{K (x)} }

where K(x 2)K (x^2) is the subfield of K(x)K (x) generated by x 2x^2, and K(x 2)K(x)¯K (x^2) \to \overline{K (x)} is algebraic, yet there is no homomorphism K(x)K(x 2)K (x) \to K (x^2) making both evident triangles commute.



Last revised on November 6, 2023 at 02:53:33. See the history of this page for a list of all contributions to it.