For a given finite extension K over \(\mathbb {Q}\) , let L/K be a finite Galois extension with Galois group G. Then, by the normal basis theorem, there exists \(\alpha \in L\) such that \(L = K[G] \cdot \alpha \) , where K[G] is the group ring. Such an element \(\alpha \) is called a normal basis generator. We say \(\alpha \in L\) is a completely normal basis generator, if \(\alpha \) is a normal basis generator for L/F for every intermediate field F such that \(K\subset F\subset L\) . In this article, we prove the following result. Let L/K be a finite Galois extension of number fields with Galois group G. If \(L\subset \mathbb {R}\) and G is an abelian group or dihedral group, then there exists a Pisot–Vijayaraghavan number \(\alpha \in L\) such that for any natural number m, \(\alpha ^m\) is a primitive element as well as a completely normal basis generator of L/K. As an application of our result, we prove the following upper bound for the index, when \(L = K\) and \(K = \mathbb {Q}\) in the above result, to get \( [\mathcal {O}_K: \mathbb {Z}[G]\cdot \alpha ] \le \left( \tau ^{n-1}|d_K|^{\frac{1}{2} - \frac{1}{2n}}+1\right) ^{n} \) where \(n = [K:\mathbb {Q}]\) , \(\tau =\phi (n)n >1\) when G is abelian and \(\tau = n+1\) when G is dihedral group of order n. We also classify the extension of prime degree related to this. We use resolvents and technique from geometry of numbers.
