## 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

### Keywords:

number field extension; discriminant; upper bound

Zbl 0827.11069
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