Originally Posted by

**Swlabr** The one to concentrate on will be multiplicative inverses.

What is the inverse of $\displaystyle \alpha+\beta \sqrt{2}$ with $\displaystyle \alpha$ and $\displaystyle \beta$ rationals?

What if both were integers, does that imply that the inverse has integer coefficients? (Alternatively, think about what the problem is saying...take $\displaystyle \beta = 0$. Then if your ring was a field every element in this subset has an inverse in the ring. This subset is just the integers. So, every integer has an inverse of the form $\displaystyle \alpha + \beta \sqrt{2}$. That is, every rational number of the form $\displaystyle \frac{1}{\gamma}, \gamma \in \mathbb{Z} \setminus \{0\}$ can be written as $\displaystyle \alpha + \beta \sqrt{2}, \alpha, \beta \in \mathbb{Z}$. As we are, allegedly, in a field we have that $\displaystyle \frac{n}{\gamma}$ is also of this form, with $\displaystyle n \in \mathbb{N}$. Thus, every rational number is of the form $\displaystyle \alpha + \beta \sqrt{2}$. This is silly!)