field extension

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.

• 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.

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

• base field
• extension field

• extension
• extension field

• 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

• extension field