field extension

English

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 $\psi$ 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:
  • $L$ may be said to be an extension field (or simply an extension) 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 intermediate field (of $L/K$), or an intermediate extension or subextension (of $K$, or perhaps of $L/K$).
  • The field $L$ is a $K$-vector space. Its dimension is called the degree of the extension, denoted $[L:K]$.
  • The construction $L/L$ 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.

Hyponyms

• algebraic extension
• transcendental extension
• simple extension
• trivial extension

Meronyms

• base field
• extension field

Related terms

• extension
• extension field

Translations

pair of fields such that one is a subfield of the other

• Finnish: kuntalaajennus

Further reading

• Field theory on Wikipedia.Wikipedia
• Glossary of field theory on Wikipedia.Wikipedia
• Tower of fields on Wikipedia.Wikipedia
• Primary extension on Wikipedia.Wikipedia
• Regular extension on Wikipedia.Wikipedia
• Extension Field on Wolfram MathWorld
• Extension of a field on Encyclopedia of Mathematics

Anagrams

• extension field