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%%% Primary Title: extension field
%%% Primary Category Code: 12F99
%%% Filename: ExtensionField.tex
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%%% Owner: drini
%%% Author(s): pahio, drini, djao
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\begin{document}

 We say that a \htmladdnormallink{field}{http://planetmath.org/encyclopedia/Field.html} $K$ is an \emph{extension} of $F$ if $F$ is a \htmladdnormallink{subfield}{http://planetmath.org/encyclopedia/Subfield.html} of $K$.

We usually denote $K$ being an extension of $F$ by \,$F\subset K$, \,$F\le K$, \,$K/F$\, or
$$\begin{xy}
*!C\xybox{
\xymatrix{
K \ar@{-}[d] \\
F
} }\end{xy}$$
One may speak of the {\em field extension} $K/F$ and call $F$ the {\em base field}.

If $K$ is an extension of $F$, we can regard $K$ as a \htmladdnormallink{vector space}{http://planetmath.org/encyclopedia/LinearSpace.html} over $F$. The \htmladdnormallink{dimension}{http://planetmath.org/encyclopedia/InfiniteDimensional.html} of this space (which could possibly be infinite) is denoted $[K:F]$, and called the {\em degree} of the extension.\footnote{ The \htmladdnormallink{term}{http://planetmath.org/encyclopedia/LeadingCoefficient.html} ``degree'' \htmladdnormallink{reflects}{http://planetmath.org/encyclopedia/Rotate.html} the fact that, in the more general setting of \htmladdnormallink{Dedekind domains}{http://planetmath.org/encyclopedia/DedekindDomain.html} and scheme-theoretic \htmladdnormallink{algebraic}{http://planetmath.org/encyclopedia/Transcendental.html} \htmladdnormallink{curves}{http://planetmath.org/encyclopedia/LocalMultiplicity.html}, the degree of an extension of \htmladdnormallink{function fields}{http://planetmath.org/encyclopedia/GenusOfAFunctionField.html} equals the algebraic degree of the \htmladdnormallink{polynomial}{http://planetmath.org/encyclopedia/LeadingCoefficient.html} defining the \htmladdnormallink{projection map}{http://planetmath.org/encyclopedia/ProjectionMap.html} of the underlying curves.}

One of the classic theorems on extensions states that if \,$F\subset K\subset L$, \,then
$$[L:F]=[L:K][K:F]$$
(in other words, degrees are \htmladdnormallink{multiplicative}{http://planetmath.org/encyclopedia/CompletelyMultiplicativeFunction.html} in towers).

\end{document}