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On Quadratic Number Fields Each Having an Unramified Extension Which Properly Contains the Hilbert Class Field of its Genus Field
Abstract
The purpose of this article is to describe simple ways to construct quadratic number fields each having an unramified extension which properly contains the Hilbert class field of its genus field (in the wide sense). The motivation of this study is the author’s observation that under the Generalized Riemann Hypothesis (GRH), for most quadratic number fields of small conductors, their maximal unramified extensions coincide with the Hilbert class fields of their genus fields. More precisely, under GRH, among the 305 imaginary quadratic number fields with discriminants larger than —1000, at most 16 fields are exceptional [39], [40], and among the 1690 real quadratic number fields with discriminants less than or equal to 5565, only 4 fields are exceptional [41].
We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2, 2, 2) whose Hilbert 2-class
fields are finite.
We report on a database of field extensions of the rationals, its properties and the methods used to compute it. At the moment the database encompasses roughly 100,000 polynomials generating distinct number fields over the rationals, of degrees up to 15. It contains polynomials for all transitive permutation groups up to that degree, and even for most of the possible combinations of signature and Galois group in that range. Moreover, whenever these are known, the fields of minimal discriminant with given group and signature have been included. The database can be downloaded from www.iwr.uni-heidelberg.de/iwr/compalg/minimum/minimum.html or from www.mathematik.uni-kassel.de~malle/minimum/minimum.html and accessed via the computer algebra system Kant. One of the aims of the compilation of this database was to test the limitations of current methods for the realization of groups as Galois groups. It turned out that these methods have limitations if the signature of the resulting Galois extension is also prescribed.
In the previous paper [15], we determined the structure of the Galois groups Gal(Kur/K) of the maximal unramified extensions Kur of imaginary quadratic number fields K of conductors (Formula Presented) 1000 under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors (Formula Presented) 723) and give a table of Gal(Kur/K). We update the table (under GRH). For 19 exceptional fields K of them, we determine Gal(Kur/K). In particular, for K = Q(Formula Presented), we obtain Gal(Kur/K) (Formula Presented) S4 × C5 and Kur = K4, the fourth Hilbert class field of K. This is the first example of a number field whose class field tower has length four.
We determine the structures of the Galois groups Gal(Kur/K) of the maximal unramified extensions Kur of imaginary quadratic number fields K of conductors ≦ 420 ≦ 719 under the Generalized Riemann Hypothesis). For all such K, Kur is K, the Hilbert class field of K, the second Hilbert class field of K, or the third Hilbert class field of K. The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We also use class number relations and a computer for calculation of class numbers of fields of low degrees in order to get class numbers of fields of higher degrees. Results on class field towers and the knowledge of the 2-groups of orders ≦ 2⁶ and linear groups over finite fields are also used.
On the experimental verification of the artin conjecture for 2-dimensional odd galois representations over Q liftings of 2-dimensional projective galois representations over Q.- A table of A5-fields.- A. Geometrical construction of 2-dimensional galois representations of A5-type. B. On the realisation of the groups PSL2(1) as galois groups over number fields by means of l-torsion points of elliptic curves.- Universal Fourier expansions of modular forms.- The hecke operators on the cusp forms of ?0(N) with nebentype.- Examples of 2-dimensional, odd galois representations of A5-type over ? satisfying the Artin conjecture.
We determine all the dihedral CM fields with relative class number one, then all of them with class number one : there are 32 such non-abelian fields with class number one. This is the first example of resolution of the class number one problem for non-abelian normal CM-fields of a given Galois group.
Let D be the discriminant of an algebraic number field F of degree n over the rational field R. The problem of finding the lowest absolute value of D as F varies over all fields of degree n with a given number of real (and consequently of imaginary) conjugate fields has not yet been solved in general. The only precise results so far given are those for n = 2, 3 and 4. The case n = 2 is trivial; n = 3 was solved in 1896 by Furtwangler, and n = 4 in 1929 by J. Mayer [6]. Reference to Furtwangler's work is given hi Mayer's paper. In this paper the results for n = 5, that is, for quintic fields, are obtained.
The Galois group of the maximal unramified p-extension (p 2) of an imaginary quadratic field is investigated for the case where the group is finite. It is shown that the group can be generated by not more than two generators with two relations. One of the relations can be taken from the 3rd term of the Zassenhaus filtration of the free group, and the second, from the 2nd, 5th, or 7th term.
Let k be a number field with Sk, the Sylow 2-subgroup of its ideal class group, isomorphic to the four-group. Then either the class number of the Hilbert class field to k is odd, or there is a unique nonabelian unramified extension L of k of degree 8. The galois group is then the dihedral or quaternion group of order 8, and the occurrence of each is characterized in terms of Hilbert's theorem 94. In the case , m a positive square-free integer, we obtain this characterization in terms of arithmetic properties of the integer m.
We compute a polynomial with Galois group SL2(11) over Q. Furthermore we prove that SL2(11) is the Galois group of a regular extension ofQ (t).
We reconstruct Brumer's family with 3-parameters of curves of genus two whose jacobian varieties admit a real multiplication of discriminant 5. Our method is based on the descent theory in geometric Galois theory which can be compared with a classical problem of Noether. Namely, we first construct a 3-parameter family of polynomials f(X) of degree 6 whose Galois group is isomorphic to the alternating group . Then we study the family of curves defined by , showing that they are equivalent to Brumer's family. The real multiplication will be described in three distinct ways, i.e., by Humbert's modular equation, by Poncelet's pentagon, and by algebraic correspondences.
Improvement factors in hazard rate and age for a sequential
preventative maintenance (PM) policy are introduced. Two imperfect PM
models are analyzed: (1) PM reduces the hazard rate while it increases
with the number of PMs, and (2) PM reduces the age. The PM is done at
intervals x k ( k =1,2, . . . , N )
and is imperfect. The optimal policies to minimize the mean cost rates
are discussed. The optimal PM sequences { x k} are
computed for a Weibull distribution
A Database for Polynomials over the Rationals
- J Klüners
- G Malle
Lösung der Aufgabe 208
- A Scholz
The class number one problem for dihedral CM-fields, in “Algebraic number theory and Diophantine analysis
- Y Lefeuvre
- S Louboutin
Lefeuvre, Corps diédraux à multiplication comlexe principaux
Some examples of unramified extensions over quadratic number fields, Sci. Rep. Tokyo Woman’s Christian Univ., Nos
- T Kondo