Splitting field

Let $F$ be a field and let $f(x)\in F[x]$ be a nonconstant polynomial.

Definition (splitting field). A field extension $E/F$ is a splitting field of $f$ over $F$ if:

1. $f$ splits over $E$, i.e. in $E[x]$ one can write $f(x)=c\prod_{i=1}^n (x-\alpha_i) \quad \text{with } c\in E^\times,\ \alpha_i\in E;$
2. $E$ is generated over $F$ by the roots of $f$: if $\alpha_1,\dots,\alpha_n$ are the roots of $f$ in $E$ (with repetition allowed), then $E = F(\alpha_1,\dots,\alpha_n),$ i.e. $E$ is obtained by adjoining those roots via a tower of fields of simple extensions .

Equivalently, after choosing an algebraic closure $\overline F$ and viewing the roots of $f$ inside $\overline F$, the splitting field is the smallest subfield of $\overline F$ containing $F$ and all roots of $f$.

A splitting field is always an algebraic extension of $F$, and if $f$ is separable then the splitting field is a separable extension . Existence and uniqueness up to $F$-isomorphism are addressed in existence/uniqueness of splitting fields and uniqueness of splitting fields .

Examples.

1. Over $F=\mathbb{Q}$, the polynomial $f(x)=x^2-2$ splits in $\mathbb{Q}(\sqrt{2})$ as $(x-\sqrt2)(x+\sqrt2)$. Since $\mathbb{Q}(\sqrt2)$ is generated by the roots, it is the splitting field.
2. Over $F=\mathbb{Q}$, the polynomial $f(x)=x^3-2$ has roots $\sqrt[3]{2},\ \zeta_3\sqrt[3]{2},\ \zeta_3^2\sqrt[3]{2}$, where $\zeta_3$ is a primitive cube root of unity . Its splitting field is $\mathbb{Q}(\sqrt[3]{2},\zeta_3)$.
3. Over $F=\mathbb{R}$, the polynomial $x^2+1$ does not split in $\mathbb{R}$ but splits in $\mathbb{C}$. Since $\mathbb{C}=\mathbb{R}(i)$ is generated by the roots $\pm i$, it is the splitting field.