Remarks

1) For each $i \in I$ let $c_i$ be a non-zero element of $K$ and let $f'_i = c_i f_i$. Then it is clear that every splitting extension for the family $(f_i)_i$, is also a splitting extension for the family $(f'_i)_{i \in I}$ and conversely. In particular, in studying splitting extensions we may limit ourselves to the case of monic polynomials.

2) Suppose that I is finite and put $f = \prod_{i \in I} f_i$. Using the uniqueness of the decomposition of a polynomial into irreducible factors in $E[X]$ (IV, p. 13, Prop. 13), we can easily show that a splitting extension for the polynomial $f$ is a splitting extension for the family $(f_i)_i$, and conversely. In other words, the case of a finite family may be reduced to the case of a single polynomial.

3) Let $f \in K[X]$ be a polynomial of degree $\geq 1$ and let $E$ be a splitting extension of $f$. If $x_1, \ldots, x_n$ are the roots off in $E$, we thus have $E = K(x_1, \ldots, x_n)$ and $[E : K]$ is finite ($V$, p. 18, Th. 2); but it may happen that $E$ is distinct from the subfields $K(x_1), \ldots, K(x_n)$ generated by a single root; this can happen even when $f$ is irreducible$^1$. We note however that when $f$ is irreducible, the fields $K(x_i)$ all have the same degree $n$ over $K$, and whenever $E$ is equal to one of them, we have $[E : K] = n$ and hence $E = K(x_1) = \cdots = K(x_n)$.