splitting fieldSplittingField2002-01-05 01:48:562002-11-25 19:01:13Definition\usepackage{amssymb}
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\usepackage{xypic}Let $f \in F[x]$ be a polynomial over a field $F$. A {\em splitting field} for $f$ is a field extension $K$ of $F$ such that
\begin{enumerate}
\item $f$ splits (factors into a product of linear factors) in $K[x]$,
\item $K$ is the smallest field with this property (any sub-extension field of $K$ which satisfies the first property is equal to $K$).
\end{enumerate}
{\bf Theorem:} Any polynomial over any field has a splitting field, and any two such splitting fields are isomorphic. A splitting field is always a normal extension of the ground field.