Splitting Field: Existence and Uniqueness

Theorem.
Let $K$ be a field and $f(x)\in K[x]$ a nonzero polynomial. There exists a field extension $L/K$ such that $f$ factors in $L[x]$ as a product of linear factors and $L$ is generated over $K$ by the roots of $f$. Such an $L$ is a splitting field of $f$ over $K$.

Moreover, if $L$ and $L'$ are splitting fields of $f$ over $K$, then there is a $K$-isomorphism $L \cong_K L'$ (not canonical). See also uniqueness of splitting fields .

Examples.

1. Over $K=\mathbb{Q}$, $f=x^2-2$ has splitting field $L=\mathbb{Q}(\sqrt2)$.
2. Over $\mathbb{Q}$, $f=x^3-2$ has splitting field $L=\mathbb{Q}(\sqrt[3]{2},\omega)$ where $\omega$ is a primitive cube root of unity.
3. Over $\mathbb{F}_2$, $f=x^2+x+1$ is irreducible and its splitting field is $\mathbb{F}_4$.

Related. polynomial rings , field extensions , normality via splitting fields .