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extension field

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extension field

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extension field

名詞

extension field (複数形 extension fields)

(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, Oxford University Press, 2006, Paperback, page 215,
  Suppose F {\displaystyle F} $F$ is a subfield of the field K {\displaystyle K} $K$ . Then K {\displaystyle K} $K$ is called an extension field of F {\displaystyle F} $F$ . So, for instance, G F ( 4 ) {\displaystyle GF(4)} $GF(4)$ and G F ( 8 ) {\displaystyle GF(8)} $GF(8)$ are extension fields of G F ( 2 ) {\displaystyle GF(2)} $GF(2)$ , although G F ( 8 ) {\displaystyle GF(8)} $GF(8)$ is not an extension field of G F ( 4 ) {\displaystyle GF(4)} $GF(4)$ .
- 1995, Terence Jackson, From Polynomials to Sums of Squares, Taylor & Francis, page 56,
  This extension field of F {\displaystyle F} $F$ always contains a root of f {\displaystyle f} $f$ in the sense that if K = F [ x ] / ( f ( x ) ) {\displaystyle K=F[x]/(f(x))} $K=F[x]/(f(x))$ then x {\displaystyle x} $x$ is a root of f ( y ) {\displaystyle f(y)} $f(y)$ in K [ y ] {\displaystyle K[y]} $K[y]$ . 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 F {\displaystyle F} $F$ , is automatically a vector space over F {\displaystyle F} $F$ . 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 F {\displaystyle F} $F$ : we say that K = F ( α ) {\displaystyle K=F(\alpha )} $K=F(\alpha )$ if K {\displaystyle K} $K$ is the field consisting of all rational expressions formed using α {\displaystyle \alpha } $\alpha$ and elements of F {\displaystyle F} $F$ .