Remark 3

Let K be a field and $\Omega$ an algebraic closure of $K$. Then the following construction may be given for a splitting extension for a family $(f_i)_{i \in I}$ of non-constant polynomials in $K[X]$: let $R_i$ be the set of roots of $f_i$ in $\Omega$ and let $R = \bigcup_{i \in I} R_i$. Then $K(R)$ is the unique subextension of $\Omega$ which is a splitting extension for $(f_i)_{i \in I}$ (V, p. 22, Prop. 5).