# Difference between revisions of "Field extension"

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If <math>L/K</math> is a field extension, then <math>L</math> may be thought of as a [[vector space]] over <math>K</math>. The dimension of this vector space is called the ''degree'' of the extension, and is denoted by <math>[L:K]</math>. | If <math>L/K</math> is a field extension, then <math>L</math> may be thought of as a [[vector space]] over <math>K</math>. The dimension of this vector space is called the ''degree'' of the extension, and is denoted by <math>[L:K]</math>. | ||

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+ | Given three fields <math>K\subseteq L\subseteq M</math>, then, if the degrees of the extensions <math>M/L</math>, <math>L/K</math> and <math>M/K</math>, are finite, then are related by the [[tower law]]: <cmath>[M:K] = [M:L]\cdot[L:M]</cmath> | ||

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## Revision as of 21:58, 1 September 2008

If and are fields and , then is said to be a **field extension**. We sometimes say that is a field extension of .

If is a field extension, then may be thought of as a vector space over . The dimension of this vector space is called the *degree* of the extension, and is denoted by .

Given three fields , then, if the degrees of the extensions , and , are finite, then are related by the tower law:

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