Let $R$ be a ring with unity, and let $\alpha$ be a ring automorphism of $R$. Now let us define a ring $R[x,\alpha]$ as follows:
1) Its addition group is $(R[x], +)$
2) Multiplication is defined like this:
$$\left(\sum_{i}^{}a_ix^i\right)\left(\sum_{j}^{}b_jx^j\right) := \sum_{i,j}^{}a_ix^ib_jx^j = \sum_{i,j}^{}(a_i\alpha^i(b_j))x^{i+j} \ \ \ (a_i, b_j \in R)$$
To be more specific, let R be $\mathbb{C}$ and let $\alpha$ be the complex conjugation automorphism.
How do I prove that the ring centre of $\mathbb{C}[x, \alpha]$ is $\mathbb{R}[x^2]$, and that the quotient-ring $\mathbb{C}[x,\alpha]/(x^2+1)$ is isomorphic to $\mathbb{H}$?