Let me preface this with saying that I'm not a math student, but rather, a physics student and I'm trying to learn algebraic topology on my own.
I would like to determine the following:
Let $H_*(\mathbb{C}\mathbb{P}^1) \cong \mathbb{Z}\langle [p] \rangle \otimes \langle [\mathbb{C}\mathbb{P}^1] \rangle$ where $p$ is a point and $[p]$ and $[\mathbb{C}\mathbb{P}^1]$ are fundamental classes. Let $a,b \in H^*(\mathbb{C}\mathbb{P}^*)$ be Poincare-dual cohomology classes, respectively. Then how can I write a presentation for the cohomology ring $H^*(\mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1)$? I am struggling with where to begin with this. How should I proceed?
Also, is there an easy way to determine $H^*(\mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1 \times \ldots \times \mathbb{C}\mathbb{P}^1)$ where there are $n$ copies of $\mathbb{C}\mathbb{P}^1$?