Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 16G / Solution

Ordinal exponentiation is defined byfor limit $\lambda$. Transfinite induction gives ${\omega }^{\alpha }\ge \alpha$. Since ${\omega }^{\alpha +1}>\alpha$, there is a least ${\alpha }^{\ast }$ with ${\omega }^{{\alpha }^{\ast }}>\alpha$. If nonzero ${\alpha }^{\ast }$ were a limit, the defining supremum would imply ${\omega }^{\beta }>\alpha$ for some $\beta <{\alpha }^{\ast }$, a contradiction. Hence ${\alpha }^{\ast }$ is a successor.

For $\alpha >0$, write ${\alpha }^{\ast }=\beta +1$. Then ${\omega }^{\beta }\le \alpha <{\omega }^{\beta +1}$. There is a largest positive integer $n$ with ${\omega }^{\beta }n\le \alpha$, and ordinal division givesRepeating on $\gamma$ terminates because there is no infinite strictly decreasing sequence of ordinals, producing the Cantor normal form

For two monomials,For general $\alpha +{\alpha }^{\prime }$, compare the leading exponent ${\beta }_1^{\prime }$ with the exponents of $\alpha$: discard every trailing term of $\alpha$ having exponent below ${\beta }_1^{\prime }$, combine coefficients if the last retained exponent equals ${\beta }_1^{\prime }$, and then append all remaining terms of ${\alpha }^{\prime }$. This is its Cantor normal form.