Let $\alpha$ be a non-zero ordinal. Prove that there exists a greatest ordinal $\beta$ such that $\omega^{\beta} \leqslant \alpha$. Explain why there exists an ordinal $\gamma$ with $\omega^{\beta}+\gamma=\alpha$. Prove that $\gamma$ is unique, and that $\gamma<\alpha$.

A non-zero ordinal $\alpha$ is called decomposable if it can be written as the sum of two smaller non-zero ordinals. Deduce that if $\alpha$ is not a power of $\omega$ then $\alpha$ is decomposable.

Conversely, prove that if $\alpha$ is a power of $\omega$ then $\alpha$ is not decomposable.

[Hint: consider the cases $\alpha=\omega^{\beta}$ ( $\beta$ a successor) and $\alpha=\omega^{\beta}$ ( $\beta$ a limit) separately.]