field extension

Noun

 * 1)  Any pair of fields, denoted L/K, such that K is a subfield of L.
 * 2) * 1998,, Basic Structures of Function Field Arithmetic, Springer, Corrected 2nd Printing, page 283,
 * Note that the extension of L obtained by adjoining all division points of $$\psi$$ includes at most a finite constant field extension.
 * 1) * 2007, Pierre Antoine Grillet, Abstract Algebra, Springer, 2bd Edition, page 530,
 * A field extension of a field K is, in particular, a K-algebra. Hence any two field extensions of K have a tensor product that is a K-algebra.
 * A field extension of a field K is, in particular, a K-algebra. Hence any two field extensions of K have a tensor product that is a K-algebra.

Usage notes

 * Related terminology:
 * $$L$$ may be said to be an (or simply an ) of $$K$$.
 * If a field $$F$$ exists which is a subfield of $$L$$ and of which $$K$$ is a subfield, then we may call $$F$$ an (of $$L/K$$), or an  or  (of $$K$$, or perhaps of $$L/K$$).
 * The field $$L$$ is a $$K$$-vector space. Its dimension is called the of the extension, denoted $$[L:K]$$.
 * The construction $$L/L$$ is called the.
 * Field extensions are fundamental in algebraic number theory and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.

Translations

 * Finnish: kuntalaajennus