Existence and uniqueness of splitting fields

Every nonconstant polynomial $f\in K[X]$ has a splitting field of finite degree over $K$. It is constructed by adjoining a root of an irreducible factor and repeating over the enlarged field. Any two splitting fields of $f$ over $K$ are isomorphic by an isomorphism that fixes $K$. Inside one fixed algebraic closure, the field generated by all roots is unique as a subfield.