In P.286 of Hatcher's 'Algebraic Topology', it is stated that $\alpha_{p^i}$ is primitive in $ \bigotimes_{i\geq 0} \mathbb{Z}_p[\alpha_{p^i}]/(\alpha_{p^i}^p)$, where 'primitive' means the coproduct $\Delta$ takes $\alpha_{p^i}$ to $1 \otimes \alpha_{p^i} + \alpha_{p^i} \otimes 1$.
I don't understand why $\Delta(\alpha_{p})$ (I have taken i=1 as an example here) cannot contain terms like $\alpha_1 \otimes \alpha_1 ^{p-1}$. I know $\alpha_1^p=0$ in the algebra but we cannot take $\alpha_1$ across the tensor product to make it trivial, can we? Is there any reason why such terms cannot exist?
I apologize if this is a stupid question, for I am not entirely familiar with tensor products, thanks in advance for any help