Proposition 4

Let K be a field and $(f_i)_{i \in I}$ a family of non-constant polynomials in $K[X]$; then there exists a splitting extension for the family $(f_i)_{i \in I}$.

We may take the polynomials $f_i$ to be monic (Remark 1). Let $i \in I$ and let the degree of $f_i$ be $d_i$. By IV, p. 73, Prop. 5 there exists a commutative algebra $A$, over $K$, not reduced to 0, and elements $\xi_{i,1}, \ldots, \xi_{i,d_i}$ of $A_i$ such that :

a) the algebra $A_i$ is generated by $(\xi_{i,1}, \ldots, \xi_{i,d_i})$;

(1) Take for example $K = \mathbf{Q}$ and $f = X^3 - 2$.

b) we have $f_i(X) = \prod_{k=1}^{d_i} (X - \xi_{i,k})$ in $A_i[X]$.

Let $A$ be the tensor product of the family of algebras $(A_i)_{i \in I}$, and let $\varphi_i$ be the canonical homomorphism of $A_i$ into $A$ (III, p. 470). The algebra $A$ is then commutative and not reduced to 0; by Krull's theorem (I, p. 104), there exists thus a maximal ideal $a$ in $A$ and $E = A/a$ is an extension of the field $K$.

Denote by $\psi$ the canonical homomorphism of $A$ into $E$ and put $x_{i,k} = \psi(\varphi_i(\xi_{i,k}))$ for $i \in I$ and $l \leq k \leq d_i$. Since the algebra $A$ is generated by $\bigcup \varphi_i(A_i)$, the extension $E$ is generated by the family $(x_{i,k})$. Further, we have $i \in I$

Therefore $E$ is a splitting extension of the family $(f_i)_{i \in I}$.