extension fieldとは 意味・読み方・使い方

名詞

extension field (複数形 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, Oxford University Press, 2006, Paperback, page 215,

     Suppose ${\displaystyle F}$ is a subfield of the field ${\displaystyle K}$. Then ${\displaystyle K}$ is called an extension field of ${\displaystyle F}$. So, for instance, ${\displaystyle GF(4)}$ and ${\displaystyle GF(8)}$ are extension fields of ${\displaystyle GF(2)}$, although ${\displaystyle GF(8)}$ is not an extension field of ${\displaystyle GF(4)}$.

   • 1995, Terence Jackson, From Polynomials to Sums of Squares, Taylor & Francis, page 56,

     This extension field of ${\displaystyle F}$ always contains a root of ${\displaystyle f}$ in the sense that if ${\displaystyle K=F[x]/(f(x))}$ then ${\displaystyle x}$ is a root of ${\displaystyle f(y)}$ in ${\displaystyle K[y]}$. 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 ${\displaystyle F}$, is automatically a vector space over ${\displaystyle F}$. 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 ${\displaystyle F}$: we say that ${\displaystyle K=F(\alpha )}$ if ${\displaystyle K}$ is the field consisting of all rational expressions formed using ${\displaystyle \alpha }$ and elements of ${\displaystyle F}$.