Let $\mathbb{Q}[\alpha]=\{g(\alpha); g\in \mathbb{Q}[X]\}$ be a field extension of the rational numbers. The field extension $\mathbb{Q}[\alpha]$ is the smallest possible subring of $\mathbb{C}$ containing the element $\alpha$ and the rational numbers. It can be seen by using the undamental theorem on homomorphisms that $\mathbb{Q}[\alpha]$ is even a field itself.
While I understand the proof using the homomorphism theorem, and see that $\mathbb{Q}[\alpha]$ is a ring, I can't see why it satisfies the inverse element (for multiplication) condition needed to be a field.
In general $g(\alpha)$ can be something like $g(\alpha)=\sum_{i=0}^\infty c_i\alpha^i, c_i\in \mathbb{Q}$. So how would I construct the inverse element, for example to $\alpha$? I can only do it for the addition but not for multiplication.
