Editing Algebraic closure (section)

==Existence of an algebraic closure and splitting fields==
Let <math>S = \{ f_{\lambda} | \lambda \in \Lambda\}</math> be the set of all [[monic polynomial|monic]] [[irreducible polynomial]]s in ''K''[''x''].
For each <math>f_{\lambda} \in S</math>, introduce new variables <math>u_{\lambda,1},\ldots,u_{\lambda,d}</math> where <math>d = {\rm degree}(f_{\lambda})</math>.
Let ''R'' be the polynomial ring over ''K'' generated by <math>u_{\lambda,i}</math> for all <math>\lambda \in \Lambda</math> and all <math>i \leq {\rm degree}(f_{\lambda})</math>. Write
: <math>f_{\lambda} - \prod_{i=1}^d (x-u_{\lambda,i}) = \sum_{j=0}^{d-1} r_{\lambda,j} \cdot x^j \in R[x]</math>

with <math>r_{\lambda,j} \in R</math>.
Let  ''I'' be the [[ideal of a ring|ideal]] in ''R'' generated by the <math>r_{\lambda,j}</math>. 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''<sub>1</sub>=''R''/''M'' has the property that every polynomial <math>f_{\lambda}</math> with coefficients in ''K'' splits as the product of <math>x-(u_{\lambda,i} + M),</math> and hence has all roots in ''K''<sub>1</sub>. In the same way, an extension ''K''<sub>2</sub> of ''K''<sub>1</sub> 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''<sub>''n''</sub> with sufficiently large ''n'', and then its roots are in ''K''<sub>''n''+1</sub>, 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''.