splitting field of a polynomial set

Given one polynomial, its splitting field is the smallest extension where it factors completely into linear pieces — you adjoin exactly enough roots and nothing more. For a whole family of polynomials, you do the same thing simultaneously: build the smallest single field in which every polynomial in the family splits. It is the minimal arena large enough to hold all the roots of all the chosen polynomials at once.

Precisely, given a set F of polynomials over K, a splitting field is an extension L of K such that every f in F factors into linear factors over L, and L is generated over K by the roots of all the f in F. For a finite set this is the same as the splitting field of the single product; for an infinite family one adjoins all roots of all members. Splitting fields exist and are unique up to a K-isomorphism.

This construction is the source of normal and Galois extensions: a finite extension is normal precisely when it is the splitting field of some polynomial, and the splitting field of a separable family is a Galois extension whose group permutes the roots. Pushing the family to 'all polynomials over K' yields the algebraic closure, so splitting fields interpolate between simple root-adjunction and full algebraic closure.