What is the cohomology algebra of finite dimensional grassmannian
$$ H^*(G_k(\mathbb{R}^N);\mathbb{Z}_2)? $$
$$ H^*(G_k(\mathbb{C}^N);\mathbb{Z}_p)? $$
$$ H^*(G_k(\mathbb{H}^N);\mathbb{Z}_p)? $$
I have obtained the infinite version
$$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k], $$ $p$ prime, $$ H^*(G_k(\mathbb{C}^\infty);\mathbb{Z}_p)=\mathbb{Z}_p[c_1,\cdots,c_k], $$ $p\geq 3$ prime, $$ H^*(G_k(\mathbb{H}^\infty);\mathbb{Z}_p)=\mathbb{Z}_p[p_1,\cdots,p_k]. $$