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The divisibility of normal integral generators. (English) Zbl 0592.12008

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 \(| \Gamma | -\varepsilon\).

MSC:

11R32 Galois theory
11R20 Other abelian and metabelian extensions
11R45 Density theorems
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
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References:

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[2] Bushnell, C.J.: Norm Distribution in Galois Orbits. J. Reine Angew. Math.310, 81-97 (1979) · Zbl 0409.12010 · doi:10.1515/crll.1979.310.81
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