Definition:Splitting Field

Let $K$ be a field, and $f$ a polynomial over $K$.

Splitting Field
A splitting fieldof $f$ over $K$ is an extension $L/K$ such that:


 * $f = k (X - \alpha_1)\cdots(X-\alpha_n)$

for some $k \in K$, $\alpha_1,\ldots,\alpha_n \in L$.

We say that $f$ splits over $L$.

Minimal Splitting Field
A field $L$ is a minimal splitting field for $f$ over $K$ if $L$ is a splitting field for $f$ and no field properly contained in $L$ has this property.