Solution
= Solution
Let $\beta=a+b\alpha\in\mathcal O_L$. The nontrivial $K$-automorphism sends $\alpha$ to $-\alpha$, so
$$
\operatorname{Tr}_{L/K}(\beta)=2a\in\mathcal O_K.
$$
Also $\alpha$ is <integral>, hence $\alpha\beta$ is <integral>, and
$$
\operatorname{Tr}_{L/K}(\alpha\beta)=2b\alpha^2
=2b(4+\sqrt{17})\in\mathcal O_K.
$$
But $4+\sqrt{17}$ has norm $-1$, so it is a unit of $\mathcal O_K$. Multiplying by its inverse proves $2b\in\mathcal O_K$.