# extension field

## English

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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 ${\displaystyle F}$ 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 ${\displaystyle F}$.

#### Usage notes

• 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 ${\displaystyle L/K}$.

#### Synonyms

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