HMIC 2015 Problem 5

Let $\omega=e^{2 \pi i / 5}$ be a primitive fifth root of unity. Prove that there do not exist integers $a, b, c, d, k$ with $k>1$ such that

$$\left(a+b \omega+c \omega^{2}+d \omega^{3}\right)^{k}=1+\omega$$