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1.
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
2.
Pavel Kraemer 《Journal of Number Theory》2004,105(2):302-321
For a real abelian field with a non-cyclic Galois group of order l2, l being an odd prime, the index of the Sinnott group of circular units is computed. 相似文献
3.
Zdeněk Polický 《Journal of Number Theory》2008,128(4):1074-1090
For a compositum of quadratic fields , where d1,…,ds are square-free odd integers and , we study the group C of circular units of k. We construct a basis of C, compute the index of C in the full group of units of k and derive a lower bound for the divisibility of this index by a power of 2. These results give a lower bound for the divisibility of the class number of the maximal real subfield of k by a power of 2. 相似文献
4.
For a large class of groups G a precise congruence subgroup of the group generated by the bicyclic units of the integral group ring ZG is determined. As an application an upper bound is calculated for the index in the unit group of ZG for the group generated by the Bass cyclic units and the bicyclic units. 相似文献
5.
Louboutin Stéphane 《manuscripta mathematica》1996,91(1):343-352
Lately, I. Miyada proved that there are only finitely many imaginary abelian number fields with Galois groups of exponents
≤2 with one class in each genus. He also proved that under the assumption of the Riemann hypothesis there are exactly 301
such number fields. Here, we prove the following finiteness theorem: there are only finitely many imaginary abelian number
fields with one class in each genus. We note that our proof would make it possible to find an explict upper bound on the discriminants
of these number fields which are neither quadratic nor biquadratic bicyclic. However, we do not go into any explicit determination. 相似文献
6.
We classify the finite groups G such that the group of units of the integral group ring ZG has a subgroup of finite index which is a direct product of free-by-free groups. 相似文献
7.
J.-M. Kim, S. Bae and I.-S. Lee showed that there exists an isomorphism between the p-primary part of the ideal class group and p-primary part of the unit group modulo cyclotomic unit group in Q+(ζpn) for all sufficiently large n under some conditions. In the present paper, we shall give an analogue of their result for modular units. 相似文献
8.
Brett A. Tangedal 《Journal of Number Theory》2007,124(2):291-313
Let F be a real quadratic field and m an integral ideal of F. Two Stark units, εm,1 and εm,2, are conjectured to exist corresponding to the two different embeddings of F into R. We define new ray class invariants and associated to each class C+ of the narrow ray class group modulo m and dependent separately on the two different embeddings of F into R. These invariants are defined as a product of special values of the double sine function in a compact and canonical form using a continued fraction approach due to Zagier and Hayes. We prove that both Stark units εm,1 and εm,2, assuming they exist, can be expressed simultaneously and symmetrically in terms of and , thus giving a canonical expression for every existent Stark unit over F as a product of double sine function values. We prove that Stark units do exist as predicted in certain special cases. 相似文献
9.
Akira Endô 《Journal of Number Theory》2011,131(10):1824-1826
Remark on Polický?s paper on circular units of a compositum of quadratic number fields is given. 相似文献
10.
We generalize Bilharz's Theorem for to all one-dimensional tori over global function fields of finite constant field. As an application, we also derive an analogue,
in the setting of function fields, of a theorem (Chen-Kitaoka-Yu, Roskam) on the distribution of fundamental units modulo
primes.
Received: 16 October 2000 / Published online: 2 December 2002
Research partially supported by National Science Council, Rep. of China. 相似文献
11.
For an abelian number field k, let CS(k) be the group of circular units of k defined by Sinnott, and CW(k) be that suggested by Washington. In this paper, we construct an element in CW(k) for a real subfield k of conductor . We will see that the order of in the factor group CW(k)/CS(k) can be very large. As an application, we derive some information about the class number of k for special cases. 相似文献
12.
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
13.
In this paper, we give parametric families of both real and complex quadratic number fields whose class group has 3-rank at least 2. As a consequence, we obtain that for all large positive real numbers x, the number of both real and complex quadratic fields whose class group has 3-rank at least 2 and absolute value of the discriminant ?x is >cx1/3, where c is some positive constant. 相似文献
14.
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
15.
Hassan Oukhaba 《Mathematische Annalen》2006,336(3):639-657
In the first part of this paper we give a new definition of the elliptic analogue of Sinnott’s group of circular units. In this we essentially use the ideas discussed in Oukhaba (in Ann Inst Fourier, 55(33):753–772, 2005). In the second part of the paper we are interested in computing the index of this group of elliptic units. This question is closely related to the behaviour of the universal signed ordinary distributions introduced in loc. cit. Such distributions have a natural resolution discovered by Anderson. Consequently, we can apply Ouyang’s general index formula and the powerful Anderson’s theory of double complex to make the computations 相似文献
16.
Norisato Kataoka 《Journal of Number Theory》2003,101(2):349-375
Let k be a real quadratic number field and the ring of integers and the group of units in k. Denote by the subgroup represented by elements of E of for a prime ideal in k. We show that for a given positive rational integer a, the set of prime numbers p for which the residual index of for lying above p is equal to a has a natural density c under the Generalized Riemann Hypothesis. Moreover, we give the explicit formula of c and conditions to c=0. 相似文献
17.
Soogil Seo 《Journal of Number Theory》2005,115(2):348-359
In his paper (Invent. Math. 109 (1992) 329-350), Solomon finds an information on the prime factorization of an element coming from a circular unit 1-ζ over the ideal class group of a real abelian number field L, where ζ denotes a root of unity. Using this he obtains an annihilator of the p-Sylow subgroup of the subgroup of the ideal class group of L generated by the classes of prime ideals lying above p. We generalize this result to the circular distributions which has the axiomatic definition of Euler systems as its defining property. 相似文献
18.
Cristian D. Popescu 《Journal of Number Theory》2005,113(2):276-307
We prove a strong form of the Brumer-Stark Conjecture and, as a consequence, a strong form of Rubin's integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum kp∞?kp·k∞ of the maximal pro-p abelian extension kp/k and the maximal constant field extension k∞/k of k, which happens to sit inside the maximal abelian extension kab of k with a quasi-finite index. This way, we extend the results obtained by the present author in (Comp. Math. 116 (1999) 321-367). 相似文献
19.
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
20.
Cristian D. Popescu 《Journal of Number Theory》2005,115(1):27-44
We show that, for all characteristic p global fields k and natural numbers n coprime to the order of the non-p-part of the Picard group Pic0(k) of k, there exists an abelian extension L/k whose local degree at every prime of k is equal to n. This answers in the affirmative in this context a question recently posed by Kisilevsky and Sonn. As a consequence, we show that, for all n and k as above, the n-torsion subgroup Brn(k) of the Brauer group Br(k) of k is algebraic, answering a question of Aldjaeff and Sonn in this context. 相似文献