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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.

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Depends. If your fields are finite, then there are well studied instances, where the products are very simple in the sense that they require very few terms. If that is your interest, then search for optimal normal bases. The cases where the normal basis consists of roots of unity of a relatively small order are particularly attractive. Such bases are used in e.g. cryptographical implementations of large finite fields. See Handbook of Cryptography for implementation instructions. If your interest is more general fields, then I cannot say much. The roots of unity still work well, though. –  Jyrki Lahtonen May 20 '12 at 12:27

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