Proposition 5

— *Let K be a field, $(f_i)_{i \in I}$ a family of non-constant polynomials in $K[X]$, E an extension of K, and F, F' subextensions of E which are each a splitting extension of $(f_i)_{i \in I}$. Then $F = F'$.

Let $R_i$ be the set of roots of $f_i$ in E and $R = \bigcup_{i \in I} R_i$. Since $f_i$ is a product of polynomials of the first degree lying in $F[X]$, we have $R_i \subset F$. By Def. 2, we have $F = K(R)$; in the same way we find that $F = K(R)$.