# extension field

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### 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, Oxford University Press, 2006, Paperback, 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)