field extension
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English[edit]
Noun[edit]
field extension (plural field extensions)
- (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 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.
- 1974, Thomas W. Hungerford, Algebra, Springer, page 230,
Usage notes[edit]
- Related terminology:
- may be said to be an extension field (or simply an extension) of .
- If a field exists which is a subfield of and of which is a subfield, then we may call an intermediate field (of ), or an intermediate extension or subextension (of , or perhaps of ).
- The field is a -vector space. Its dimension is called the degree of the extension, denoted .
- The construction 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.
Related terms[edit]
Translations[edit]
pair of fields such that one is a subfield of the other
Further reading[edit]
- 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