I have a question about Galois groups of reducible separable polynomials. For example x^5+x+1 = (x^2+x+1)(x^3-x^2+1). We can see by direct computation that it's solvable and not contained in A5. The first factor has Gal = Z/2, and the other one, according to the program GAP, has Gal of order 2, so S2. My question then, what is Gal for x^5+x+1, is it S2 x S2? If so, is it the case for f with factorization p1...p_n and G:=Gal(p_i) that Gal(f) = G1 x ... x G_n (nope)? Thank you.
edit: I might just throw in this stupid question as well: If all the factors of f=p1...p_n has solvable Galois group, is f also solvable?