Ideal class groups of cyclotomic number fields III

Abstract

[255] c Instytut Matematyczny PAN, 2008 256 F. LemmermeyerIn this article we will show that k{(p)} contains K = Q(ζ p(p−1) ), and that k{(p)}/K is abelian and unramified. In particular we will see that Scholz's construction gives a subfield of the Hilbert class field of K.Classically, proofs that certain extensions are unramified were often done by applying Abhyankar's lemma, which gives sufficient conditions for killing tame ramification in extension fields:Lemma 2 (Abhyankar's lemma). Let L 1 /K and L 2 /K be finite extensions of algebraic number fields, and let L = L 1 L 2 denote their compositum. For a prime ideal P in L, let e j (j = 1, 2) denote the ramification indices of the primeAbhyankar's lemma can be used to construct unramified extensions quite easily; its disadvantage lies in the fact that it cannot handle wild ramification. We will circumvent this problem by describing the extensions via ideal groups and using basic class field theory. Our main result is the construction of class fields corresponding to various factors of class numbers of certain cyclotomic number fields that have been found over the last fourty years. Such factors have been constructed using Remarks. 1. The negative solution of the class field tower problem by Golod and Shafarevich implies the existence of number fields with arbitrarily large degree and bounded root discriminant, hence lim inf n→∞ E n /n = 0 (this follows already from Scholz's results given above). Scholz's original conjecture that lim n→∞ E n /n = 0 seems to be still open.2. The best upper bounds for lim inf E n used to come from examples due to Martinet, whose records have recently been improved by Hajir and Maire [9,10].3. Scholz communicated most of the results in [22] to Hasse in a letter from Aug. 22, 1936 (see [14]). In this letter he wrote he doubted that E p = O(p/log p) where p runs through the primes.4. There are a lot of open questions regarding the behaviour of root discriminants. The following problem is particularly appealing and might well be accessible with the tools we know today. Let us call a group G metabelian of level m if the mth derived group G (m) is trivial but G (m−1) = 1. Also let log r denote the rth iterated logarithm, i.e., log 0 n = n, log 1 n = log n, and log r+1 n = log log r n. Ankeny [1] showed that there is a constant Ideal class groups 257 c > 0 such that log rd(K) > c log m (K : Q) for all normal extensions K/Q whose Galois group G is metabelian of level m ≥ 1. It seems not to be known whether this is best possible in the following sense: for any m ≥ 1, does there exist a sequence of metabelian extensions K/Q of level m such that log rd(K) = O(log m (K : Q))? The answer to this question is clearly positive for m = 1, where the abelian extensions K = Q(ζ p ) satisfyScholz's results imply that the answer is also positive for m = 2 since log rd(K) < 2 log 2 (K : Q) for the metabelian extensions he constructed.10. Scholz's construction. We start by introducing the notation and by explaining the relevant facts from class fi...

Der Briefwechsel Hasse - Scholz - Taussky

“…Die Scholzsche Konstruktion liefert tatsächlich unverzweigte abelsche Erweiterungen geeigneter Kreiskörper; vgl. Lemmermeyer [168]. Die Scholzsche Vermutung lim n→∞ E n /n = 0 ist trotz der Techniken von Golod und Shafarevich immer noch offen.…”

“…Eine Lücke in Furtwänglers Beweis wurde später durch diesen selbst, sowie durch Scholz [S6] geschlossen. 4 Wie in [168] gezeigt wird, werden die Scholzschen Strahlklassenkörperüber geeigneten Erweiterungen zu Hilbertklassenkörpern. Insbesondere sind Scholz' Strahlklassenkörper modulo p von Q(ζp−1) Hilbertklassenkörperüber Q(ζ p(p−1) ).…”

“…[168]. Sei En = | disc K| 1/n die " Wurzeldiskriminante" des Zahlkörpers K vom Grad n mit minimaler Diskriminante disc K. Aus der Existenz unendlicher Klassenkörpertürme folgt sofort lim inf En = 0.…”

On Hasse's Unit Index

“…Our proof relies on computing the distribution of the 4-rank of narrow class groups Cl + (8d) of the real quadratic fields Q( √ 2d) for 2d ∈ D 2 . For 2d ∈ D 2 , the 4-rank of these groups turns out to be substantially larger on average than the 4-rank of narrow class groups of real quadratic fields; compare [2, (7), p. 458] to (5.1). As a result, unlike in the case of generic real quadratic fields, it is not possible to deduce a positive proportion of 2d ∈ D 2 with 4-rank of Cl + (8d) equal to 0 by simply studying the first moment of the 4-rank.…”