Let $L/K$ be a finite Galois extension. The normal basis theorem says that there is an element $a\in L$ such that $\{ g(a) | g\in \text{Gal}(L/K)\}$ is a basis of $L$ as a $K$-vector space.
Let $g,g'\in \text{Gal}(L/K)$. Expressing the element $g(a)+g'(a)$ with respect to this basis is trivial, but expressing the element $g(a)g'(a)\in L$ seems hard to me. Does anyone know how to do this?
A followup question (in the case where the above is not generally easy) might be: any basis is well-adapted to sums of its elements; are there other bases which are also well-adapted to products? "Well-adapted" here means one could actually carry out computations and prove things; e.g. products are also messy when considering the basis $\{1,\alpha,\alpha^2,...,\alpha^{\deg ({m_{\alpha,K}})-1}\}$ of $L=K(\alpha)$; you'll have the coefficients of the minimal polynomial $m_{\alpha,K}$ playing an important role.