x^{5}
= y + y^{5} = y(1 + y^{4}), and since 1 + y^{4} is always
positive

it follows that x and y cannot have opposite signs.
Using the other three equations similarly, it follows that
x, y, z and t must all have the same signs or all be zero.

Adding the four equations gives x + y + z + t = 0, so they
cannot all have the same signs, and so must all be zero:
x = y = z = 0.