Extension field

English

Noun

1. Template:Lb A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements.
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Usage notes

• Not to be confused with Template:M, 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 Template:M.
  • 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 Template:M and can be regarded as a vector space of dimension 1.
• An extension field of degree 2 (respectively, 3) may be called a Template:M (respectively, Template:M).
• A field $F$ which is both a subfield of $L$ and an extension field of $K$ may be called an Template:M, Template:M or Template:M of the field extension $L/K$.

Synonyms

• Template:Sense Template:L Template:Qualifier

Hyponyms

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Related terms

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Further reading

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• Extension Field on Template:W
• Extension of a field on Template:W

Anagrams

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