# field extension

## English

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### Noun

field extension (plural field extensions)

1. (algebra, field theory, algebraic geometry) Any pair of fields, denoted L/K, such that K is a subfield of L.
• 1974, Thomas W. Hungerford, Algebra, Springer, page 230:
A Galois field extension may be defined in terms of its Galois group (Section 2) or in terms of the internal structure of the extension (Section 3).
• 1998, David Goss, 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 ${\displaystyle \psi }$ includes at most a finite constant field extension.
• 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.

#### Usage notes

• Related terminology:
• ${\displaystyle L}$ may be said to be an extension field (or simply an extension) of ${\displaystyle K}$.
• If a field ${\displaystyle F}$ exists which is a subfield of ${\displaystyle L}$ and of which ${\displaystyle K}$ is a subfield, then we may call ${\displaystyle F}$ an intermediate field (of ${\displaystyle L/K}$), or an intermediate extension or subextension (of ${\displaystyle K}$, or perhaps of ${\displaystyle L/K}$).
• The field ${\displaystyle L}$ is a ${\displaystyle K}$-vector space. Its dimension is called the degree of the extension, denoted ${\displaystyle [L:K]}$.
• The construction ${\displaystyle L/L}$ is called the trivial extension.
• 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.