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The Maximal Unramified Extensions of the Imaginary Quadratic Number Fields with Class Number Two
Abstract
The maximal unramified extensions of the imaginary quadratic number fields with class number two are determined explicitly under the Generalized Riemann Hypothesis.
... Remark 1.3. The above obstructions for realizing M(q 2 ) are neccessary but not sufficient, for example, from [21] one can derive that there are no M(q 2 )-extensions when K is a totally imaginary quadratic number field with class number 2. ...
... It is not true, however, that Γ ur K has to be solvable (see for example [9]). In [21] Yakamura determines the unramified Galois group for all imaginary quadratic fields K of class number 2 and shows that in this situation Γ ur K is finite. The results of [21] are unconditional, except for the case Q( √ −427) where the generalized Riemann hypothesis is assumed. ...
... In [21] Yakamura determines the unramified Galois group for all imaginary quadratic fields K of class number 2 and shows that in this situation Γ ur K is finite. The results of [21] are unconditional, except for the case Q( √ −427) where the generalized Riemann hypothesis is assumed. Yakamura uses discriminant bounds to determine Γ ur K and as such, his methods are of a of quantitative nature. ...
We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply this obstruction theory to give an infinite family of groups together with an infinite family of number fields such that for any number field K in this family, cannot be realized as an unramified Galois group over K. To prove this result, we determine the ring structure of the \'etale cohomology ring where is the ring of integers of an arbitrary totally imaginary number field $K.
... 1. $h(K)=2$ . [9] . , . ...
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.
All algebraic number fields F of degree 5 and absolute discriminant less than 2 × 107 (totally real fields), respectively 5 × 106 (other signatures) are determined. We describe the methods which we applied and list significant data.
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.
Vita. Thesis--University of Illinois. Includes bibliographical references. Photocopy of typescript.
In this paper, we determine all the imaginary abelian number fields with class number one. There exist exactly 172 imaginary abelian number fields with class number one. The maximal conductor of these fields is 10921 = 67 · 163, which is the conductor of the biquadratic number field .
Seit Herr Artin seine allgemeinen L -Funktionen, die mit Frobeniusschen Gruppencharakteren gebildet sind, entdeckt hat [1], sind die multiplikativen Relationen Dedekindscher ζ -Funktionen als additive Relationen zwischen den Frobeniusschen Gruppencharakteren mit Erfolg untersucht worden. Durch diese Methode sind gewisse Klassenzahlrelationen in den folgenden Zeilen zu betrachten.
Nombre de classes d'ide ? aux d'une extension die ? drale d'un corps de nombres
- C Castela
C. Castela, Nombre de classes d'ide ? aux d'une extension die ? drale d'un corps de nombres, C.R. Acad. Sci. Paris Ser. A 287 (1978), 483?486
- S Kuroda
- Ber Die Klassenzahlen Algebraischer Zahlko Rper
S. Kuroda, U ber die Klassenzahlen algebraischer Zahlko rper, Nagoya Math. 1 (1950),
1110.
- G Frey
- W D Geyer
- U Ber Die Fundamentalgruppe Von Ko Rpern Mit Divisorentheorie
- J Reine Angew
G. Frey und W. D. Geyer, U ber die Fundamentalgruppe von Ko rpern mit Divisorentheorie, J. Reine Angew. Math. 254 (1972), 1100122.