I am a beginner trying to understand Dan Quillen's idempotent map $\xi:MU\_{(p)} \to MU\_{(p)}$ described in his classic paper 5-page paper [On formal group laws...](http://www.ams.org/journals/bull/1969-75-06/S0002-9904-1969-12401-8/S0002-9904-1969-12401-8.pdf)


Let $F$ be a formal group law over a ring $R$ and and $f(x)\in R[[x]$ with $f(0)=0$. He defines $V\_nf$ and $F\_nf$ to be elements of $R[[x]$ defined by
$$(V\_nf)(x)=f(x^n)$$
$$(F\_nf)(x)={\sum\_{i=1}^n}^F f(\zeta^iX^{1/n}) $$
where the sum above uses $F$. Now fix a prime $p$.
Given a formal groups law $F$, he defines a change of variables $$c\_F^{-1}(x)=\sum\_{(n,p)=1} \frac{\mu(n)}{n}V\_nF\_n \gamma\_0$$ where $\gamma\_0(x)=x$ that makes $(c\_{F\*}F)(x,y):=c\_FF(c\_F^{-1}x, c\_F^{-1}y)$ into a p-typical formal group law. 
My first question is can someone show the elementary algebra that establishes that $c\_Fx$ is a p-typical group law.


Now let $F$ denote the formal group law for $MU\_{(p)}$, and let $\xi=c\_FF$ (Quillen writes $C\_F\Omega$ and I think this is what he means... or maybe he means $\xi=c\_FF\_{MU\_{(p)}})$. So $\xi$ is a p-typical group law over $MU^\*\_{(p)}$, and we know $MU\_\*$ is universal for formal group laws. So we have a map of rings $\xi:MU\_\* \to MU^\*\_{(p)}$ and hence $\xi:MU^\*\_{(p)} \to MU^\*\_{(p)}$. Okay, I understand that. Quillen then claims that for every $X$ we have a map $\hat{\xi}:MU^\*\_{(p)}(X) \to MU^\*\_{(p)}(X)$...and my question is why? Edit: that has been answered by Dylan Wilson in the comments as the map $\xi$ tensored with $MU^\*(X)$, since we have $MU^\*(X)\_{(p)}=MU^\*(X) \otimes\_{MU\_\*} MU^\*\_{(p)}$.



very basic question on p-typical group law and MU and BP
For the first question: In the universal (torsion-free) case, this can be found in Ravenel's book "Complex Cobordism and
Stable Homotopy Groups of Spheres" starting on page 345. [The book](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel.pdf) is available online on its author's website.


