extension field

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, 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 ${\displaystyle F}$ .

• Not to be confused with field extension, which refers to the construction ${\displaystyle L/K}$
• The extension field ${\displaystyle L}$ constitutes a vector space over ${\displaystyle K}$ (i.e., a ${\displaystyle K}$ -vector space).
  • A minimal set ${\displaystyle B}$ comprising one element of ${\displaystyle K}$ plus additional elements not in ${\displaystyle K}$ which together generate ${\displaystyle L}$ is called a basis.
  • The dimension of the vector space (aka the degree of the extension), is denoted ${\displaystyle [L:K]}$ and is equal to the cardinality of ${\displaystyle B}$ .
• In the case ${\displaystyle L=K}$ , ${\displaystyle L}$ 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 ${\displaystyle F}$ which is both a subfield of ${\displaystyle L}$ and an extension field of ${\displaystyle K}$ may be called an intermediate field, intermediate extension or subextension of the field extension ${\displaystyle L/K}$ .

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

• number field
• splitting field

• field extension

• Field extension on Wikipedia.Wikipedia
• 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

• field extension