extension field

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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[edit]

  • 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 .


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


Related terms[edit]


Further reading[edit]