I am trying to understand part of Milnor-Novikov theorem about multiplicative generators of $MU_* \cong \mathbb Z[x_1, x_2, \dots]$ using S.Kochman’s “Bordsim, Stable Homotopy and Adams Spectral Sequences”.

Namely, define \begin{equation} \gamma_n = \begin{cases} p\quad&\text{if $n = p^t -1$ for some prime $p$ and positive integer $t$}; \\ 1\quad&\text{otherwise}. \end{cases} \end{equation}

Theorem. (Proposition 4.3.16 in the book) Element $x_n \in MU_*$ is a polynomial generator iff \begin{equation} h(x_n) = \pm\gamma_n A_n + \mathrm{decomposable}\in H_*(MU; \mathbb Z). \end{equation}

I have understood the case $n \neq p^t-1$. Can anybody explain to me what happens next? Why does $h(x_{p^t-1})$ divisible by $p$ and not divisible by and other prime $p’$?

It seems that this is because of bidegree of $h(x_{p^t-1})$ in ASS, but I cannot understand why.

May be you can suggest me any readable source (different from Stong’s or Ravenel’s books) to find the proof?


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