Take the tour ×  Let $\Gamma_R[\alpha]$ denote the divided
polynomial algebra over $R$; that is, the quotient of the free
$R$-algebra $R\langle \alpha_1,\alpha_2,\cdots \rangle$ by the
relations $$\alpha_n \cdot \alpha_m =
\binom{n+m}{n}\alpha_{n+m},$$ with $t_0=1$. I am particularly
interested in the case where $R = \mathbb{F}_p$.  The claim is
that $$\Gamma_{\mathbb{F}_p}[\alpha] \simeq \bigotimes_{k \ge 0}
\mathbb{F}_p[\alpha_{p^i}]/(\alpha_{p^i}^p)$$  (Note: I presume
the tensor product is over $\mathbb{F}_p$ here?)  The proof is
given is Hatcher's algebraic topology book pp. 286-287. I can
understand what he is doing (sort-of), but I can't see how it
all combines to give a proof.  First he claims that
$\Gamma_{\mathbb{F}_p}[\alpha] =
\Gamma_{\mathbb{Z}}[\alpha]\otimes \mathbb{F}_p$. I don't see
why this is the case?  Then he claims that this is equivalent to
the statement  $\ast$ The element
$\alpha_1^{n_0}\alpha_{p}^{n_1} \cdots \alpha_{p^k}^{n_k}$ in
$\Gamma_\mathbb{Z}[\alpha]$ is divisible by $p$ iff