In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits, i.e., decomposes into linear factors. Wikipedia

Given a polynomial p over a field K, the smallest extension field L of K such that p, as a polynomial over L, decomposes into linear factors (polynomials of degree 1); (of a set of polynomials) given a set P of polynomials over K, the smallest extension field of K over which every polynomial in P decomposes into linear factors. Wiktionary