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 Kalgebra. Hence any two field extensions of K have a tensor product that is a Kalgebra.
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.
Hyponyms[edit]
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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