The divisibility of normal integral generators (Q1075371)

Let \(K\) be a finite Galois extension of \(\mathbb Q\), with Galois group \(\Gamma\) of odd prime order. Let \(a\) be an element of \(K\) with \(K=a\cdot\mathbb Q\Gamma\) and let \(T\) be any finite set of rational primes. The author shows that for any Euclidean norm \(\|\cdot \|\) on \(\mathbb R\Gamma\) and any \(\varepsilon >0\), there is a constant \(c>0\) (depending on the norm, and on \(a\), \(T\) and \(\varepsilon)\), such that \[ | N_{K/\mathbb Q}(a\cdot x)|_ T>c\| x\|^{1-\varepsilon} \] for all \(x\in \mathbb Z\Gamma\). The author remarks that he expects the exponent in the bound to be improved to \(| \Gamma | -\varepsilon\).