Bases of number fields with small height
Let \(K\) be a number field of degree \(d\) over \({\mathbb Q}\). Let \(\{a_1,\ldots,a_d\}\) be a basis of \(K\) viewed as a vector space over \({\mathbb Q}\). Let us define its height as the relative projective height \(H_K(a_1:\cdots a_d)\). Among the bases of \(K\), there is one of smallest height; let us denote this smallest height by \(B(K)\). It is known [\textit{J. H. Silverman}, Duke Math. J. 51, 395-403 (1984; Zbl 0579.14035)] that, for all number fields, \(B(K)\geq d^{-d/2}| D_K|^{1/2}\), where \(D_K\) denotes the discriminant of \(K\). In the present paper it is proved that for totally real number fields also \[ B(K)\geq C_1(d)| D_K|^{1/2}, \] where \(C_1(d)\) can be taken to be \(2^{d(3d+1)/2}\). In the general case it is proved that for every \(\varepsilon>0\), \(B(K)\ll_{d,\varepsilon} R(K)^{1-\varepsilon} | D_K|^{1/2+\varepsilon}\), where the involved constant are ineffective.
bases of number fields with small height
scientific article; zbMATH DE number 1026405
scientific article