scholarly journals The class groups of the imaginary Abelian number fields with Galois group (ℤ/2ℤ)n

2004 ◽  
Vol 70 (2) ◽  
pp. 267-277 ◽  
Author(s):  
Jeoung-Hwan Ahn ◽  
Soun-Hi Kwon
Keyword(s):  
Galois Group ◽  
Riemann Hypothesis ◽  
Number Fields ◽  
Class Groups

Assuming the Generalised Riemann Hypothesis we determine all imaginary Abelian number fields N whose Galois group G(N/ℚ) is isomorphic to (ℤ/2ℤ)n for some integers n ≥ 1 and the square of every ideal of N is principal.

scholarly journals Fitting Ideals in Number Theory and Arithmetic

2021 ◽  
Author(s):  
Cornelius Greither
Keyword(s):  
Number Theory ◽  
Galois Group ◽  
Classical Result ◽  
Number Fields ◽  
Class Groups ◽  
Stark Conjecture

AbstractWe describe classical and recent results concerning the structure of class groups of number fields as modules over the Galois group. When presenting more modern developments, we can only hint at the much broader context and the very powerful general techniques that are involved, but we endeavour to give complete statements or at least examples where feasible. The timeline goes from a classical result proved in 1890 (Stickelberger’s Theorem) to a recent (2020) breakthrough: the proof of the Brumer-Stark conjecture by Dasgupta and Kakde.

scholarly journals Logarithmic derivatives of Artin -functions

2013 ◽  
Vol 149 (4) ◽  
pp. 568-586 ◽  
Author(s):  
Peter J. Cho ◽  
Henry H. Kim
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Lower Bounds ◽  
Riemann Hypothesis ◽  
Number Fields ◽  
Upper And Lower Bounds ◽  
Zero Density ◽  
Galois Closures ◽  
Derivatives Of ◽  
Logarithmic Derivatives

AbstractLet $K$ be a number field of degree $n$, and let $d_K$ be its discriminant. Then, under the Artin conjecture, the generalized Riemann hypothesis and a certain zero-density hypothesis, we show that the upper and lower bounds of the logarithmic derivatives of Artin $L$-functions attached to $K$ at $s=1$ are $\log \log |d_K|$ and $-(n-1) \log \log |d_K|$, respectively. Unconditionally, we show that there are infinitely many number fields with the extreme logarithmic derivatives; they are families of number fields whose Galois closures have the Galois group $C_n$ for $n=2,3,4,6$, $D_n$ for $n=3,4,5$, $S_4$ or $A_5$.

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

scholarly journals Explicit methods for the Hasse norm principle and applications to A n and S n extensions

2021 ◽  
pp. 1-41
Author(s):  
ANDRÉ MACEDO ◽  
RACHEL NEWTON
Keyword(s):  
Galois Group ◽  
Computational Methods ◽  
Number Fields ◽  
Normal Closure ◽  
Explicit Methods ◽  
Weak Approximation ◽  
Norm Principle ◽  
Theoretical Results

Abstract Let K/k be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for K/k and the defect of weak approximation for the norm one torus \[R_{K/k}^1{\mathbb{G}_m}\] . We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of K/k has symmetric or alternating Galois group.

The p -Dimension of Class Groups of Number Fields

1970 ◽  
Vol 2 (Part_3) ◽  
pp. 525-529 ◽  
Author(s):  
I. Connell ◽  
D. Sussman
Keyword(s):  
Number Fields ◽  
Class Groups

scholarly journals A Note on Ideal Class Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 239-247 ◽  
Author(s):  
Kenkichi Iwasawa
Keyword(s):  
Class Group ◽  
Cyclotomic Field ◽  
Theoretical Method ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ideal Class Groups ◽  
The Ideal

In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.

scholarly journals Correction to: Stickelberger ideals and Fitting ideals of class groups for abelian number fields

2019 ◽  
Vol 374 (3-4) ◽  
pp. 2083-2088
Author(s):  
Masato Kurihara ◽  
Takashi Miura
Keyword(s):  
Number Fields ◽  
Class Groups
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