Show that, for any nonconstant , there is a ﬁeld K containing F such that p(x) splits over K.

I know that if q(x) is any nonconstant polynomial with coeﬃcients from F , then there a ﬁeld L containing F such that there is a root of q(x) in L.

Now suppose p(x) is the product of irreducible polynomials then we can take to be the field in which has a root.

My question is : would the direct sum of these fields be a field over which p(x) would split (supposing that in , splits )?