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Definition: Let f be a polynomial with coefficients in a field F, not necessarily an irreducible polynomial. A splitting field for f over F is an extension field K/F such that f splits completely in K, say f(x)=(x-a_1)(x-a_2)...(x-a_n) with a_i in K, and K is generated by the roots: K=F(a_1, a_2, ... , a_n).
Source: Algebra, second edition by Michael Artin