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The number of extensions of a number field with fixed degree and bounded discriminant. (English) Zbl 1099.11068

Let \(L/K\) denote an extension of the number field \(K\) of degree \([L:K]=n\). For a fixed \(K\), denote by \(N_{K,n}(X)\) the number of fields \(L\) (up to \(K\)-isomorphism) such that and the norm \(N_ Q^K\) of \({\mathcal D}_{L/K}\), the relative discriminant of \(L/K\), is less than \(X\). It is a folk conjecture that \(N_{K,n}(X)\sim c_{K,n}X\) where \(n\) is fixed and \(X\) tends to infinity. For \(n\leq 2\) this is trivial, for \(n=3\) the conjecture is known to be true due to work of Davenport and Heilbronn (\(K=\mathbb Q\)) and Datskovsky and Wright in general. Recently Bhargava dealt with \(n=4,5\). For general \(n\) the best upper bound has been \(N_{K,n}(X)\ll X^{(n+2)/4}\) with the implied constants depending upon \(K\) and \(n\). This was proved by W. M. Schmidt [Columbia University number theory seminar, New York, 1992. Paris: Société Mathématique de France, Astérisque 228, 189–195 (1995; Zbl 0827.11069)]. The authors greatly improve on this by showing that, for all \(n>2\), \[ N_{K,n}(X)\ll(X{\mathcal D}_K^n A_n^{[K:Q]})^{{\exp} (C\sqrt{\log n})}, \] where \({\mathcal D}_K\) is the absolute value of the discriminant of \(K\), \(A_n\) is a constant depending only on \(n\), and \(C\) is an absolute constant. The main idea in Schmidt’s proof is to count monic polynomials in \({\mathbb Z}[x]\) whose coefficients have (real) absolute values bounded in terms of \(N_Q^K({\mathcal D}_{L/K})\). Indeed, every extension \(L/K\) contains an integer \(\alpha\) which is a zero of such a polynomial. The present authors’ idea is to consider \(r\)-tuples of such integers instead of single integers. As they write themselves, their proof requires only elementary arguments from the geometry of numbers and linear algebra. Furthermore, the authors obtain a lower bound \(\gg_K X^{1/2+1/{n^2}}\), \(n>2\), for the number of fields \(L\) satisfying the above conditions together with the condition that the Galois closure of \(L\) has Galois group \(S_n\) over \(K\) (thus establishing a major improvement of an earlier result due to G. Malle). Finally they obtain the upper bound \[ N_{K,n}(X;\text{ Gal}) \ll_{K,n,\varepsilon}X^{3/8+\varepsilon}, \] for the number \(N_{K,n}(X;\text{ Gal})\) of Galois extensions \(L/K\) among those counted by \(N_{K,n}(X)\) in case \(n>4\).

MSC:

11R45 Density theorems
11R29 Class numbers, class groups, discriminants

Citations:

Zbl 0827.11069
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