Definition 2

Let K be a field and $(f_i)_{i \in I}$ a family of non-constant polynomials in $K[X]$. By a splitting extension of $(f_i)_i$, we understand any extension $E$ of $K$ with the following properties :

a) For each $i \in I$, the polynomial $f_i$ splits in $E[X]$ into a product of polynomials of degree 1.

b) For each $i \in I$ let $R_i$ be the set of roots of $f_i$ in $E$, then $E = K \left( \bigcup_{i \in I} R_i \right)$.

Sometimes the term « splitting field » is used instead of « splitting extension ».