共查询到20条相似文献,搜索用时 15 毫秒
1.
J. E. Carter 《Archiv der Mathematik》2003,81(3):266-271
Let K be a number field and let G be a finite abelian group. We call
K a Hilbert-Speiser field of type G if, and only if, every tamely
ramified normal extension L/K with Galois group isomorphic to G has
a normal integral basis. Now let C2 and C3 denote the cyclic
groups of order 2 and 3, respectively. Firstly, we show that among all
imaginary quadratic fields, there are exactly three Hilbert-Speiser
fields of type $C_{2}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-1, -3, -7\}$.
Secondly, we give some necessary and sufficient conditions for a real
quadratic field $K = \mathbb{Q}(\sqrt {m})$ to be a Hilbert-Speiser field of
type C2. These conditions are in terms of the congruence class of m
modulo 4 or 8, the fundamental unit of K, and the class number of K.
Finally, we show that among all quadratic number fields, there are
exactly eight Hilbert-Speiser fields of type $C_{3}: \mathbb{Q}(\sqrt {m})$,
where $m \in \{-11,-3, -2, 2, 5, 17, 41, 89\}$.Received: 2 April 2002 相似文献
2.
The purpose of this paper is to exhibit a new family of real bicyclic biquadratic fields K for which we can write the Hasse unit index of the group generated by the units of the three quadratic subfields in the unit
group E
K
of K. As a byproduct, one can explicitly relate the class number of K with the product of the class numbers of the three quadratic subfields.
Received: 25 July 2000 / Revised version: 12 December 2000 相似文献
3.
We show that if L/
K is a degree p
extension of number fields which is wildly ramified at a prime ${\frak p}$ of
K of residue characteristic
p, then the ramification groups of ${\frak p}$
(in the splitting field of L over
K) are uniquely determined by the
${\frak p}$-adic valuation of the discriminant of L
/K.Received: 3 July 2002 相似文献
4.
Wolfgang M. Ruppert 《manuscripta mathematica》1998,96(1):17-22
If K is a number field of degree n over Q with discriminant D
K
and if α∈K generates K, i.e. K=Q(α), then the height of α satisfies with . The paper deals with the existence of small generators of number fields in this sense. We show: (1) For each $n$ there are
infinitely many number fields K of degree $n$ with a generator α such that . (2) There is a constant d
2 such that every imaginary quadratic number field has a generator α which satisfies .?(3) If K is a totally real
number field of prime degree n then one can find an integral generator α with .
Received: 10 January 1997 / Revised version: 13 January 1998 相似文献
5.
For an algebraic number field k and a prime number p (if p=2, we assume that μ4⊂k), we study the maximal rank ρ
p
of a free pro-p-extension of k. This problem is related to deep conjectures of Greenberg in Iwasawa theory. We give different equivalent formulations of
these conjectures and we apply them to show that, essentially, ρ
k
=r
2(k)+1 if and only if k is a so-called p-rational field.
Received: 29 April 1999 / Revised version: 31 January 2000 相似文献
6.
Sey Kim 《Journal of Number Theory》2006,121(1):7-29
Given any distinct prime numbers p,q, and r satisfying certain simple congruence conditions, we display a congruence relation between the fundamental units for the biquadratic field , modulo a certain prime ideal of OK. This congruence in particular implies the validity of the equivariant Tamagawa number conjecture formulated by Burns and Flach for the pair (h0(SpecK),Z[Gal(K/Q)]). 相似文献
7.
On the distribution of irreducible algebraic integers 总被引:1,自引:0,他引:1
Jerzy Kaczorowski 《Monatshefte für Mathematik》2009,156(1):47-71
We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic
number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate
for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this
is always the case when h = 2.
Author’s address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań,
Poland
The author was supported in part by KBN Grant # N N201 1482 33. 相似文献
8.
Hiroki Sumida 《manuscripta mathematica》1999,98(4):477-490
Let p be a prime number, K a finite abelian extension of Q containing p-th roots of unity and K
n
the n-th layer of the cyclotomic Z
p
-extension of K. Under some conditions we construct an element of K
n
from an ideal class of the maximal real subfield of K
n
. We determine whether its p-th root is contained by some Z
p
-extension of K
n
or not for each n, using the zero of p-adic L-function and the order of the ideal class group of the maximal real subfield of K
m
for sufficiently large m.
Received: 13 February 1998 / Revised version: 30 September 1998 相似文献
9.
A. Mouhib 《Journal of Number Theory》2009,129(6):1205-1211
This paper investigates the 2-class group of real multiquadratic number fields. Let p1,p2,…,pn be distinct primes and . We draw a list of all fields K whose 2-class group is trivial. 相似文献
10.
Stéphane Louboutin 《manuscripta mathematica》2001,106(4):411-427
Let \lcub;K
m
}
m
≥ 4 be the family of non-normal totally real cubic number fields associated with the Q-irreducible cubic polynomials P
m
(x) =x
3−mx
2−(m+1)x− 1, m≥ 4. We determine all these K
m
's with class numbers h
m
≤ 3: there are 14 such K
m
's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents
e
m
of the ideal class groups of these K
m
go to infinity with m and we determine all these K
m
's with ideal class groups of exponents e
m
≤ 3: there are 6 suchK
m
with ideal class groups of exponent 2, and 6 such K
m
with ideal class groups of exponent 3.
Received: 16 November 2000 / Revised version: 16 May 2001 相似文献
11.
Humio Ichimura 《Journal of Number Theory》2008,128(4):858-864
Let p be a prime number. We say that a number field F satisfies the condition when for any cyclic extension N/F of degree p, the ring of p-integers of N has a normal integral basis over . It is known that F=Q satisfies for any p. It is also known that when p?19, any subfield F of Q(ζp) satisfies . In this paper, we prove that when p?23, an imaginary subfield F of Q(ζp) satisfies if and only if and p=43, 67 or 163 (under GRH). For a real subfield F of Q(ζp) with F≠Q, we give a corresponding but weaker assertion to the effect that it quite rarely satisfies . 相似文献
12.
Let K/F be a Kummer cyclic extension of number fields. In the case when the degree is a prime number, Gómez Ayala gave an explicit criterion for the existence of a normal integral basis. More recently Ichimura proposed a generalization of that result for cyclic extensions of arbitrary degree, but we have found that Ichimura’s result is incorrect. In this paper we present a counter-example to Ichimura’s result as well as the correct generalization of Gómez Ayala’s result. 相似文献
13.
Guy Henniart 《Mathematische Annalen》2001,319(1):75-87
Résumé. Soit L/K une extension galoisienne finie de corps p-adiques, et soit F un corps de nombres totalement réel, dont le complété en une place v est isomorphe à {\it K}. Si p est impair, nous montrons qu'il existe une extension galoisienne finie E de F, totalement réelle, de même degré que L sur K, et dont le complété en v est isomorphe àL ; quand p vaut 2, nous prouvons une version affaiblie. Ces résultats interviennent dans une preuve des conjectures de Langlands pour
sur les corps p-adiques.
Received March 27, 2000 / Revised May 8, 2000 / Published online December 8, 2000 相似文献
14.
Xia Gao 《Journal of Number Theory》2011,131(6):994-1019
Text
The goal of this note is to generalize a formula of Datskovsky and Wright on the zeta function associated with integral binary cubic forms. We show that for a fixed number field K of degree d, the zeta function associated with decomposable forms belonging to K in d−1 variables can be factored into a product of Riemann and Dedekind zeta functions in a similar fashion. We establish a one-to-one correspondence between the pure module classes of rank d−1 of K and the integral ideals of width <d−1. This reduces the problem to counting integral ideals of a special type, which can be solved using a tailored Moebius inversion argument. As a by-product, we obtain a characterization of the conductor ideals for orders of number fields.Video
For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=RePyaF8vDnE. 相似文献15.
Let F be a cubic cyclic field with exactly one ramified prime p,p>7, or , a real quadratic field with . In this paper, we study the 3-primary part of K2OF. If 3 does not divide the class number of F, we get some results about the 9-rank of K2OF. In particular, in the case of a cubic cyclic field F with only one ramified prime p>7, we prove that four conclusions concerning the 3-primary part of K2OF, obtained by J. Browkin by numerical computations for primes p, 7≤p≤5000, are true in general. 相似文献
16.
Ludwig Gauckler 《Archiv der Mathematik》2008,90(2):136-139
Let p be a rational prime and let a be an integer which is divisible by p exactly to the first power. Then the Galois group of the Eisenstein polynomial f = X
p
+ aX + a is known to be either the full symmetric group S
p
or the affine group A(1, p), and it is conjectured that always G = S
p
. In this note we settle this conjecture for p = 5 and, answering a question by J.-P. Serre, we show that this does not carry over when replacing the integer a by some rational number with 5-adic valuation equal to 1.
Received: 6 June 2007 相似文献
17.
Alexis Michel 《Mathematische Zeitschrift》1999,230(3):451-470
Let E be an elliptic curve with complex multiplication over the ring of integers of an imaginary quadratic field K. Denote by p an odd prime that splits into in and by the unique -extension of K totally ramified above . It is well-known that the Selmer group attached to any finite extension of is analogous to the minus part of the p-class group of divisors of the cyclotomic - extensions of CM number fields. One of the most striking examples of this analogy is the existence of a translation formula à la Kida for the codimension of the Selmer group at the top of the tower. In this article we carry on the analogy with the presentations
of results similar to those proven by Gold and Madan in the cyclotomic case (see [8]), which were the continuation of Kida's
work. More precisely, we describe the -structure of the Selmer group when G is a cyclic group of order p or . In addition, we study the modular representation of G on the subgroup of points of order p of the Selmer group, when G is cyclic of order .
Received December 3, 1997 相似文献
18.
Manfred Kolster 《Mathematische Annalen》2002,323(4):667-692
Let E be a totally complex abelian number field with maximal real subfield F, and let denote the non-trivial character of . Similar to the classical case n=1 the value of the Artin L-function at for odd is given by a relative class number formula of the form
Here is a higher Q-index, which is equal to 1 or 2 and is a higher relative class number. Here for any number field L the higher class number is the order of the finite group
closely related to the order of the higher K-theory group of the ring of integers in L.
Received: 4 June 1999 / Revised version: 27 September 2001 / Published online: 26 April 2002 相似文献
19.
In this paper abelian function fields are restricted to the subfields of cyclotomic function fields. For any abelian function field K/k with conductor an irreducible polynomial over a finite field of odd characteristic, we give a calculating formula of the relative divisor class number of K. And using the given calculating formula we obtain a criterion for checking whether or not the relative divisor class number is divisible by the characteristic of k. 相似文献
20.
We prove that the submodule in K-theory which gives the exact value
of the L-function by the Beilinson regulator map at non-critical values for Hecke characters of imaginary quadratic fields K with cl (K) = 1(p-local Tamagawa number conjecture) satisfies that the length of its coimage under the local Soulé regulator map is the p-adic valuation of certain special values of p-adic L-functions associated to the Hecke characters. This result yields immediately, up to Jannsens conjecture, an upper bound for
in terms of the valuation of these p-adic L-functions, where Vp denotes the p-adic realization of a Hecke motive.Received: 4 June 2003 相似文献