Splitting field

Definition. Let $f(x)\in K[x]$ be a nonzero polynomial. A splitting field of $f$ over $K$ is a field extension $L/K$ such that:

1. $f$ factors in $L[x]$ as a product of linear factors, and
2. $L$ is generated over $K$ by the roots of $f$ (equivalently, $L$ is minimal with property (1)).

Splitting fields are unique up to $K$-isomorphism (see existence and uniqueness of splitting fields ).

See also. normal extension , Galois extension , polynomial ring .

Examples.

1. For $f(x)=x^2-2\in\mathbb{Q}[x]$, the splitting field is $\mathbb{Q}(\sqrt2)$.
2. For $f(x)=x^3-2\in\mathbb{Q}[x]$, the splitting field is $\mathbb{Q}(\sqrt[3]{2},\zeta_3)$, where $\zeta_3$ is a primitive cube root of unity.
3. For $f(x)=x^2+1\in\mathbb{R}[x]$, the splitting field over $\mathbb{R}$ is $\mathbb{C}$.