Splitting polynomials and fields: Definitions

Let F be any field, and f be a monic polynomial of degree n in F[X]. This polynomial is said to split in F if it factors completely, i.e., factors as a product of n linear factors x-ri. The ri are then the roots of f, that is, the solutions of the equation f(x)=0. If K is some extension of F, we likewise say f splits in K if can be written as a product (x-r1)(x-r2)...(x-rn) of n linear factors in K[X]. Clearly f then splits also in F(r1,r2,...rn), the subfield of K generated by the roots. We say that K is a splitting field of f over F if f splits in K and K=F(r1,r2,...rn).