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?

It's true, if you are talking about ideals.
Clearly, $(2(15+4\sqrt{14})) \subset (2)$. For the other inclusion, note that $(15+4\sqrt{14})(15-4\sqrt{14}) = 1$.
