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We prove that for any Hecke eigenform f of level 1 and arbitrary weight thereis a self-dual cuspidal automorphic form $\pi$ of $GL_6(\Q)$ corresponding to$\Symm^5 (f)$, i.e., such that the system of Galois representations attached to$\pi$ agrees with the 5-th symmetric power of the one attached to f. We alsoimprove the base change result that we obtained in a previous work: for anynewform f, and any totally real number field F (no extra assumptions on f orF), we prove the existence of base change relative to the extension $F/\Q$.Finally, we combine the previous results to deduce that base change also holdsfor $\Symm^5(f)$: for any Hecke eigenform f of level 1 and any totally realnumber field F, the automorphic form corresponding to $\Symm^5 (f)$ can be basechanged to F.

Automorphy of Symm^5(GL(2)) and base change