Field extension

Revision as of 21:58, 1 September 2008 by Jam (talk | contribs) (tower law)

If $K$ and $L$ are fields and $K\subseteq L$, then $L/K$ is said to be a field extension. We sometimes say that $L$ is a field extension of $K$.

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

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

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