Number fields of given degree and bounded discriminant. (English) Zbl 0827.11069

Schmidt, Wolfgang M.

Columbia University number theory seminar, New York, 1992. Paris: Société Mathématique de France, Astérisque. 228, 189-195 (1995).

It is shown that the number \(N(d, x)\) of algebraic number fields \(K\) of given degree \(d\) and of discriminant \(d(K)\leq x\) satisfies \(N(d,x)= O(x^{(d+ 2)/4})\). It is known that for \(d= 2,3\) one has \(N(d, x)= (c_d+ o(1))x\) with a certain non-zero \(c_d\) [H. Davenport and H. Heilbronn, Proc. R. Soc. Lond., Ser. A 322, 405-420 (1971; Zbl 0212.081) for \(d=3\)] and it is conjectured that a similar result holds for all \(d\geq 2\).
For the entire collection see [Zbl 0815.00008].

Reviewer: W.Narkiewicz (Wrocław)

MSC:

11R29	Class numbers, class groups, discriminants
11N45	Asymptotic results on counting functions for algebraic and topological structures
11R04	Algebraic numbers; rings of algebraic integers

Keywords:

algebraic number fields; degree; discriminant

Citations:

Zbl 0212.081