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**vincisonfire** Show that, for any nonconstant $\displaystyle p(x) \in F(x) $, 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 $\displaystyle q_i(x) $ then we can take $\displaystyle L_i = F(x)/q_i(x) $ to be the field in which $\displaystyle q_i(x) $ 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 $\displaystyle L_i $, $\displaystyle q_i(x) $ splits )?