Lemma 1 (Cauchy,Kronecker)   Let $K$ be a field. For any irreducible polynomial $f$ in $K[X]$ there is an extension field of $K$ in which $f$ has a root.

Proof. If $I$ is the ideal generated by $f$ in $K[X]$ , since $f$ is irreducible, $I$ is a maximal ideal of $K[X]$ , and consequently $K[X]/I$ is a field.
We can construct a canonical monomorphism $v$ from $K$ to $K[X]$ . By tracking back the field operation on $K[X]/I$ , $v$ can be extended to an isomorphism $w$ from an extension field $L$ of $K$ to $K[X]/I$ .
We show that $\alpha = w^{-1}(X+I)$ is a root of $f$ .
If we write $f = \sum_{i = 1}^n f_i X^i$ then $f+I = 0$ implies:
which means that $f(\alpha) = 0$ .

Theorem 1   Let $K$ be a field and let $M$ be a finite set of nonconstant polynomials in $K[X]$ . Then there exists an extension field $L$ of $K$ such that every polynomial in $M$ splits in $L[X]$

Proof. If $L$ is a field extension of $K$ then the nonconstant polynomials $f_1, f_2, ... ,f_n$ split in $L[X]$ iff the polynomial $\prod_{i=1}^n f_i$ splits in $L[X]$ . Now the proof easily follows from the above lemma.