extension field

Noun

 * 1)  A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements.

Usage notes

 * Not to be confused with, which refers to the construction $$L/K$$
 * The extension field $$L$$ constitutes a vector space over $$K$$ (i.e., a $$K$$-vector space).
 * A minimal set $$B$$ comprising one element of $$K$$ plus additional elements not in $$K$$ which together generate $$L$$ is called a.
 * The dimension of the vector space (aka the degree of the extension), is denoted $$[L:K]$$ and is equal to the cardinality of $$B$$.
 * In the case $$L=K$$, $$L$$ is called the and can be regarded as a vector space of dimension 1.
 * An extension field of degree 2 (respectively, 3) may be called a (respectively, ).
 * A field $$F$$ which is both a subfield of $$L$$ and an extension field of $$K$$ may be called an, or  of the field extension $$L/K$$.