Bug report (z[0]+z[1]+z[2])^5 == z0^5 + z1^5 + z2^5

edit

This is not a bug.

sage: var('x,y,z')
sage: ((x+y+z)^5).expand()
x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + y^5 + 5*x^4*z + 20*x^3*y*z + 30*x^2*y^2*z + 20*x*y^3*z + 5*y^4*z + 10*x^3*z^2 + 30*x^2*y*z^2 + 30*x*y^2*z^2 + 10*y^3*z^2 + 10*x^2*z^3 + 20*x*y*z^3 + 10*y^2*z^3 + 5*x*z^4 + 5*y*z^4 + z^5

Note the divisibility by 5 of all terms which are "missing" in your characteristic $p=5$ calculation.

This is a consequence of the Freshman's dream: $(x+y)^p \equiv x^p + y^p \pmod p$; just apply it twice.

It looks too good to be true, but it really is true in characteristic $p$ (the proof is in the linked article).