 Solution

If $\beta \in {\mathcal{O}}_L$, all of its $K$-conjugates are algebraic integers, so their sum and product give

Conversely, if these two quantities lie in ${\mathcal{O}}_K$, then $\beta$ is a root ofa monic polynomial whose coefficients are algebraic integers. Part (a), or transitivity of integral dependence, implies $\beta \in {\mathcal{O}}_L$.