extension field

English

Noun

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

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

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

Usage notes

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

Synonyms

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

Hyponyms

• number field
• splitting field

Related terms

• field extension

Translations

field that contains a subfield

Further reading

• 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

Anagrams

• field extension