# Definition:Splitting Field

## Definition

### Of a polynomial

Let $K$ be a field.

Let $f$ be a polynomial over $K$.

A splitting field of $f$ over $K$ is a field extension $L / K$ such that:

$f = k \paren {X - \alpha_1} \cdots \paren {X - \alpha_n}$

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

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

### Of a set of polynomials

Let $K$ be a field.

Let $\FF$ be a set of polynomials over $K$.

A splitting field of $\FF$ over $K$ is a field extension $L / K$ such that for any $f \in \FF$:

$f = k \paren {X - \alpha_1} \cdots \paren {X - \alpha_n}$

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

We say that $\FF$ splits over $L$.

## Minimal Splitting Field

### Of a polynomial

Let $K$ be a field.

Let $f$ be a polynomial over $K$.

Let $L/K$ be a field extension of $K$.

Then $L$ is a minimal splitting field for $f$ over $K$ if $L$ is a splitting field for $f$ and no field extension properly contained in $L$ has this property.

### Of a set of polynomials

Let $K$ be a field.

Let $\FF$ be a set of polynomials over $K$.

Let $L / K$ be a field extension of $K$.

Then $L$ is a minimal splitting field for $\FF$ over $K$ if $L$ is a splitting field for $f$ and no field extension properly contained in $L$ has this property.