= Solution
If $\beta\in\mathcal O_L$, all of its $K$-conjugates are algebraic integers, so their sum and product give
$$
\operatorname{Tr}_{L/K}(\beta),\ N_{L/K}(\beta)\in\mathcal O_K.
$$
Conversely, if these two quantities lie in $\mathcal O_K$, then $\beta$ is a root of
$$
X^2-\operatorname{Tr}_{L/K}(\beta)X+N_{L/K}(\beta),
$$
a monic <polynomial> whose coefficients are algebraic integers. Part (a), or transitivity of <integral> dependence, implies $\beta\in\mathcal O_L$.
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