Let S = \{ f_{\lambda} \mid \lambda \in \Lambda\} be the set of all
monic irreducible polynomials in K[x]. For each f_{\lambda} \in S, introduce new variables u_{\lambda,1},\ldots,u_{\lambda,d} where d = {\rm degree}(f_{\lambda}). Let R be the polynomial ring over K generated by u_{\lambda,i} for all \lambda \in \Lambda and all i \leq {\rm degree}(f_{\lambda}). Write : f_{\lambda} - \prod_{i=1}^d (x-u_{\lambda,i}) = \sum_{j=0}^{d-1} r_{\lambda,j} \cdot x^j \in R[x] with r_{\lambda,j} \in R. Let I be the
ideal in R generated by the r_{\lambda,j}. Since I is strictly smaller than R, Zorn's lemma implies that there exists a maximal ideal M in R that contains I. The field K_1=R/M has the property that every polynomial f_{\lambda} with coefficients in K splits as the product of x-(u_{\lambda,i} + M), and hence has all roots in K_1. In the same way, an extension K_2 of K_1 can be constructed, etc. The union of all these extensions is the algebraic closure of K, because any polynomial with coefficients in this new field has its coefficients in some K_n with sufficiently large n, and then its roots are in K_{n+1}, and hence in the union itself. It can be shown along the same lines that for any subset S of K[x], there exists a
splitting field of S over K. ==Separable closure==