
Originally Posted by
bleys
Well, for any non-zero $\displaystyle a$ in the integral domain, you must show $\displaystyle a^{-1}$ is also in the integral domain. We've shown it for $\displaystyle \alpha$, what about the rest? How is knowing $\displaystyle \alpha ^{-1}$ is in $\displaystyle K[\alpha]$ imply any polynomial in $\displaystyle \alpha$ is invertible?
I'm not really sure where you're going; the best I could do is use induction on the degree of polynomials in $\displaystyle K[\alpha]$, but I don't think that's the kind of proof you're (or the book is) thinking about, even though it is quite simple and short.