$\omega^2$ be of the form $\alpha + \omega$?

Question

Why cant $\omega^2$ be of the form $\alpha + \omega$ for any ordinal $\alpha$? I thought otherwise $\omega + \omega^2 = \omega(1 + \omega) = \omega^2 = \alpha + \omega$ therefore $\omega + \alpha + \omega = \alpha + \omega$ but i dont know if this produces a contradiction or not. Or is there another approach?

Answer 1

Lets say that $w^2=\alpha+w$ for some ordinal $\alpha$, which must be greater than $w$. By the division of ordinals it implies that $\alpha=w\beta+n$ for some $\beta$ ordinal and $n\in w$. So $$\alpha+w=w\beta+n+w=w\beta+(n+w)\\=w\beta+w=w(\beta+1)=w^2\iff \beta+1=w$$ It implies that $w$ is the succesor of $\beta$, which contradicts the fact that $w$ is a limit ordinal.