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Noun

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extension field (plural extension fields)

  1. (algebra, field theory) A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements.
    • 1992, James G. Oxley, “Matroid Theory”, in Paperback, Oxford University Press, published 2006, page 215:
      Suppose   is a subfield of the field  . Then   is called an extension field of  . So, for instance,   and   are extension fields of  , although   is not an extension field of  .
    • 1995, Terence Jackson, From Polynomials to Sums of Squares, Taylor & Francis, page 56:
      This extension field of   always contains a root of   in the sense that if   then   is a root of   in  . It then follows that any polynomial will have roots, either in the original field of its coefficients or in some extension field.
    • 1998, Neal Koblitz, Algebraic Aspects of Cryptography, Volume 3, Springer, page 53:
      An extension field, by which we mean a bigger field containing  , is automatically a vector space over  . We call it a finite extension if it is a finite vector space. By the degree of a finite extension we mean its dimension as a vector space. One common way of obtaining extension fields is to adjoin an element to  : we say that   if   is the field consisting of all rational expressions formed using   and elements of  .

Usage notes

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  • Not to be confused with field extension, which refers to the construction  
  • The extension field   constitutes a vector space over   (i.e., a  -vector space).
    • A minimal set   comprising one element of   plus additional elements not in   which together generate   is called a basis.
    • The dimension of the vector space (aka the degree of the extension), is denoted   and is equal to the cardinality of  .
  • In the case  ,   is called the trivial extension and can be regarded as a vector space of dimension 1.
  • An extension field of degree 2 (respectively, 3) may be called a quadratic extension (respectively, cubic extension).
  • A field   which is both a subfield of   and an extension field of   may be called an intermediate field, intermediate extension or subextension of the field extension  .

Synonyms

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  • (field that contains a subfield): extension (where the base field is given)

Hyponyms

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Translations

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Further reading

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Anagrams

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