Field extension

English

Noun

1. Template:Lb Any pair of Template:L, denoted L/K, such that K is a subfield of L.
   • Template:Quote-book
   • 1998, Template:W, 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 Template:M (or simply an Template:M) 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 Template:M (of $L/K$), or an Template:M or Template:M (of $K$, or perhaps of $L/K$).
  • The field $L$ is a $K$-vector space. Its dimension is called the Template:M of the extension, denoted $[L:K]$.
  • The construction $L/L$ is called the Template:M.
• 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

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Meronyms

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Related terms

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Translations

• Finnish: Template:T

Further reading

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• Extension Field on Template:W
• Extension of a field on Template:W

Anagrams

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