field adjunction

Let K be a field and E an extension fieldMathworldPlanetmath of K.  If αE, then the smallest subfieldMathworldPlanetmath of E, that contains K and α, is denoted by K(α).  We say that K(α) is obtained from the field K by adjoining the element α to K via field adjunction.

Theorem.K(α) is identical with the quotient field Q of K[α].

Proof. (1) Because K[α] is an integral domainMathworldPlanetmath (as a subring of the field E), all possible quotients of the elements of K[α] belong to E. So we have


and because K(α) was the smallest, then  K(α)Q.

(2) K(α) is a subring of E containing K and α, therefore also the whole ring K[α], that is,  K[α]K(α).  And because K(α) is a field, it must contain all possible quotients of the elements of K[α], i.e.,  QK(α).

In to the adjunction of one single element, we can adjoin to K an arbitrary subset S of E:  the resulting field K(S) is the smallest of such subfields of E, i.e. the intersection of such subfields of E, that contain both K and S.  We say that K(S) is obtained from K by adjoining the set S to it.  Naturally,


The field K(S) contains all elements of K and S, and being a field, also all such elements that can be formed via addition, subtraction, multiplication and division from the elements of K and S.  But such elements constitute a field, which therefore must be equal with K(S).  Accordingly, we have the

Theorem.K(S) constitutes of all rational expressions formed of the elements of the field K with the elements of the set S.


1. K(S) is the union of all fields K(T) where T is a finite subset of S.
2. K(S1S2)=K(S1)(S2).
3. If, especially, S also is a subfield of E, then one may denote  K(S)=KS.


  • 1 B. L. van der Waerden: AlgebraPlanetmathPlanetmath. Erster Teil.  Siebte Auflage der Modernen Algebra. Springer-Verlag; Berlin, Heidelberg, New York (1966).
Title field adjunction
Canonical name FieldAdjunction
Date of creation 2015-02-21 15:39:45
Last modified on 2015-02-21 15:39:45
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Definition
Classification msc 12F99
Synonym simple extension
Related topic GroundFieldsAndRings
Related topic Forcing
Related topic PolynomialRingOverFieldIsEuclideanDomain
Related topic AConditionOfAlgebraicExtension