field adjunction
Let $K$ be a field and $E$ an extension field^{} of $K$. If $\alpha \in E$, then the smallest subfield^{} of $E$, that contains $K$ and
$\alpha $, is denoted by $K(\alpha )$. We say that $K(\alpha )$ is
obtained from the field $K$ by adjoining the element $\alpha $
to $K$ via field adjunction.
Theorem. $K(\alpha )$ is identical with the quotient field $Q$ of $K[\alpha ]$.
Proof. (1) Because $K[\alpha ]$ is an integral domain^{} (as a subring of the field $E$), all possible quotients of the elements of $K[\alpha ]$ belong to $E$. So we have
$$K\cup \{\alpha \}\subseteq K[\alpha ]\subseteq Q\subseteq E,$$ |
and because $K(\alpha )$ was the smallest, then $K(\alpha )\subseteq Q.$
(2) $K(\alpha )$ is a subring of $E$ containing $K$ and $\alpha $, therefore also the whole ring $K[\alpha ]$, that is, $K[\alpha ]\subseteq K(\alpha )$. And because $K(\alpha )$ is a field, it must contain all possible quotients of the elements of $K[\alpha ]$, i.e., $Q\subseteq K(\alpha )$.
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,
$$K\subseteq K(S)\subseteq E.$$ |
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 $S$.
Notes.
1. $K(S)$ is the union of all fields $K(T)$ where $T$ is a finite subset of $S$.
2. $K({S}_{1}\cup {S}_{2})=K({S}_{1})({S}_{2})$.
3. If, especially, $S$ also is a subfield of $E$, then one may denote $K(S)=KS$.
References
- 1 B. L. van der Waerden: Algebra^{}. 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 |