Write down the recursive definitions of ordinal addition, multiplication and exponentiation. Show that, for any nonzero ordinal $\alpha$, there exist unique ordinals $\beta, \gamma$ and $n$ such that $\alpha=\omega^{\beta} . n+\gamma, \gamma<\omega^{\beta}$ and $0<n<\omega$.

Hence or otherwise show that $\alpha$ (that is, the set of ordinals less than $\alpha$ ) is closed under addition if and only if $\alpha=\omega^{\beta}$ for some $\beta$. Show also that an infinite ordinal $\alpha$ is closed under multiplication if and only if $\alpha=\omega^{\left(\omega^{\gamma}\right)}$ for some $\gamma$.

[You may assume the standard laws of ordinal arithmetic, and the fact that $\alpha \leqslant \omega^{\alpha}$ for all $\alpha$.]