Is it true that $(4+\sqrt{14})(4-\sqrt{14}) = (4+\sqrt{14})^2$ in $\mathbb{Q}(\sqrt{14})$? Is it true that $(4+\sqrt{14})(4-\sqrt{14}) = (4+\sqrt{14})^2$ in $\mathbb{Q}(\sqrt{14})$? I am going through the solution of a problem I'm working on and this seems to be what they are saying. If its true, why so? I see that the left hand side is $(2)$ and the right hand side is $(2(15+4\sqrt{14}))$ but then what?

No. $\mathbb{Q}(\sqrt{14})$ is an integral domain, so if we had equality it would imply that
$$4 - \sqrt{14} = 4 + \sqrt{14}$$
by cancelling a factor of $4 + \sqrt{14}$ from each side. Rearranging would lead to $2 \sqrt{14} = 0$, which is false.
