Let $Z\sim N(0,1)$. Give a clean and rigorous proof that $Z^3$ cannot be normally distributed

Question

Just as the title suggests. I simply have no idea how to prove that a random variable is not normally distributed. I'd really appreciate some hint.

Answer 1

If $Z^3$ were normally distributed, then it would have variance $\mathbf{E}(Z^6) = 15$ (https://en.wikipedia.org/wiki/Normal_distribution#Moments) but this contradicts at once the easy fact $\mathbf{P}(Z^3 \in (-1,1)) = \mathbf{P}(Z \in (-1,1)).$