$E$-(co)homology of $BU(n)$ (Reference request) I am currently reading Lurie's notes on Chromatic Homotopy Theory (252x) and in Lecture 4 (https://www.math.ias.edu/~lurie/252xnotes/Lecture4.pdf), he skims through the calculation of $E^{\ast}(BU(n))$ and $E_\ast(BU(n))$ where $E^\ast(-)$ is a complex oriented cohomology theory. I am able to follow the calculation but am struggling with filling in the gaps. Could someone kindly refer me to a resource where these computations have been explicitly carried out?
PS- I have tried having a look at Adams's blue book, I would appreciate it if there's a relatively modern reference/textbook.
 A: I prefer an approach that does not go via ordinary cohomology.  I'll assume you're happy that $E^*(BU(1))=E^*(\mathbb{C}P^\infty)=E^*[[x]]$.  We can then use a Künneth theorem to get $E^*(BU(1)^d)=E^*[[x_1,\dotsc,x_d]]$.  We have an inclusion $i\colon BU(1)^d\to BU(d)$, giving
$$ i^*\colon E^*(BU(d))\to E^*(BU(1)^d)=E^*[[x_1,\dotsc,x_d]]. $$
This is permutation-invariant up to homotopy, and so gives a map from $E^*(BU(d))$ to the subring of symmetric functions in $x_1,\dotsc,x_d$.  This subring is just $E^*[[c_{d1},\dotsc,c_{dd}]]$, where $c_{di}$ is the $i$'th elementary symmetric function in $x_1,\dotsc,x_d$.  Our task is to prove that the resulting map
$$ i^* \colon E^*(BU(d)) \to E^*[[c_{d1},\dotsc,c_{dd}]] $$
is an isomorphism.
One approach is to use the Projective Bundle Theorem, as follows.  Let $V$ be a complex bundle of dimension $d$ over a CW complex $X$.  We then have an associated projective bundle $PV$ over $X$, with fibre $P(V_a)\simeq\mathbb{C}P^{d-1}$ over each point $a\in X$.  There is then a tautological line bundle $L$ over $PV$, which is classified by a map $f\colon PV\to BU(1)$, giving an Euler class $f^*(x)\in E^2(PV)$.  We will just write $x$ instead of $f^*(x)$.  We now have a map
$$ \phi\colon\bigoplus_{i=0}^{d-1} E^{*-2i}(X) \to E^*(PV) $$
given by $\phi(a_0,\dotsc,a_{d-1})=\sum_ia_ix^i$.  More generally, we have a map $\phi_U\colon\bigoplus_iE^{*-2i}(U)\to E^*(P(V|_U))$ for each open subspace $U$ of $X$.  If $V$ can be trivialised over $U$ then $P(V|_U)\simeq U\times\mathbb{C}P^{d-1}$ and it is easy to see that $\phi_U$ is an isomorphism.  The maps $\phi_U$ are easily seen to be compatible with Mayer-Vietoris sequences, so if $\phi_U$, $\phi_W$ and $\phi_{U\cap W}$ are isomorphisms, then so is $\phi_{U\cup W}$ (by an application of the five lemma).  If $X$ is compact then it can be written as the union of finitely many open sets over which $V$ can be trivialised, and it follows that $\phi_X$ is an isomorphism.  We can now pass to limits to show that $\phi$ is an isomorphism in all cases.  In particular, this means that the map $\pi^*\colon E^*(X)\to E^*(PV)$ is injective.
Now take $V$ to be the tautological bundle over $BU(d)$, which we can think of as the Grassmannian of $d$-dimensional subspaces of $\mathbb{C}^\infty$.  From this point of view it is not hard to identify $PV$ with $BU(1)\times BU(d-1)$, and we can assume inductively that $E^*(BU(d-1))$ is as expected.  It follows that the ring
$$ E^*(BU(1)\times BU(d-1)) = E^*[[x,c_{d-1,1},\dotsc,c_{d-1,d-1}]] $$
is freely generated by $\{1,x,\dotsc,x^{d-1}\}$ as a module over $E^*(BU(d))$. On the other hand, straightforward algebra shows that it is also freely generated by the same set as a module over $E^*[[c_{d1},\dotsc,c_{dd}]]$.  The induction step can be deduced from this.
