Additive cohomology of BO(k)

Question

we know that $H^*(BO(n);\mathbb{Z}/2) \cong \mathbb{Z}/2[w_1,\cdots,w_n]$, where $w_i \in H^i(BO(n);\mathbb{Z}/2)$ is the $i$-th Stiefel Whitney class.

Is it correct to say $\langle w_i \rangle = H^i(BO(n);\mathbb{Z}/2)$?

Thanks!

Answer 1

No, this is not correct. For instance, in $H^2(BO(n);\mathbb{Z}/2)$, we have not just $w_2$, but also $w_1^2$, which is linearly independent from $w_2$ (over $\mathbb{Z}/2$). More generally, a basis for $H^i(BO(n);\mathbb{Z}/2)$ over $\mathbb{Z}/2$ is given by the set of all monomials $w_1^{m_1}w_2^{m_2}\dots w_n^{m_n}$, where $m_1,\dots, m_n$ are nonnegative integers such that $\sum_{j=1}^{n} jm_j=i$.