On the Class-number of the Maximal Real Subfield of a Cyclotomic Field

For any square-free positive integer m, let H(m) be the class-number of the field , where ζ_m is a primitive m-th root of unity. We show that if m = {3(8 g + 5)}² ? 2 is a square-free integer, where g is a positive integer, then H(4 m) > 1. Similar result holds for a square-free integer m = {3(8 g +7)}² ?2, where g is a positive integer. We also show that n|H(4 m) for certain positive integers m and n.     

On the parity of the class number of multiquadratic number fields

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.     

A note on the 3-rank of quadratic fields

The difference between the 3-rank of the ideal class group of an imaginary quadratic field and that of the associated real quadratic field is equal to 0 or 1. In this note, we give an infinite family of examples in each case.Received: 9 September 2002     

On congruence relations between the fundamental units of biquadratic fields

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)]).     

The ideal class groups of dihedral extensions over imaginary quadratic fields and the special values of the Artin L-function

We study the relation between the minus part of the p-class subgroup of a dihedral extension over an imaginary quadratic field and the special value of the Artin L-function at 0.     

On the factorization of the relative class number in terms of Frobenius divisions

The relative class number of an imaginary abelian number fieldK is—up to trivial factors—the product of the first Bernoulli numbersB _x belonging to the odd characters ofK. This product splits into rational factorsF _Z = {B ; Z}, whereZ runs through the Frobenius divisions of odd characters. It is shown that each numberF _z is—up to a certain prime power—the index of two explicitly given subgroups of (K, +). These subgroups are cyclic Galois modules, whose generators arise from roots of unity and cotangent numbers, resp. Our result is an analogue of a result concerningh ⁺ which was given by Leopoldt many years ago.To the memory of my friend Kurt Dietrich     

Hilbert-Speiser number fields for a prime p inside the p-cyclotomic field

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 .     

The 3-Sylow subgroup of the tame kernel of real number fields

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 K₂O_F. If 3 does not divide the class number of F, we get some results about the 9-rank of K₂O_F. 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 K₂O_F, obtained by J. Browkin by numerical computations for primes p, 7≤p≤5000, are true in general.     

On some questions related to the Gauss conjecture for function fields

We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.     

Continued fractions, special values of the double sine function, and Stark units over real quadratic fields

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.     

Average values of L-functions in characteristic two

Gauss made two conjectures about average values of class numbers of orders in quadratic number fields, later on proven by Lipschitz and Siegel. A version for function fields of odd characteristic was established by Hoffstein and Rosen. In this paper, we extend their results to the case of even characteristic. More precisely, we obtain formulas of average values of L-functions associated to orders in quadratic function fields over a constant field of characteristic two, and then derive formulas of average class numbers of these orders.     

Several types of algebraic numbers on the unit circle

Let A _p ⊂ C denote the set of all algebraic numbers such that α ∈ A _p if and only if α is a zero of a (not necessarily irreducible) polynomial with positive rational coefficients. We give several results concerning the numbers in A _p . In particular, the intersection of A _p and the unit circle |z| = 1 is investigated in detail. So we determine all numbers of degree less than 6 on the unit circle which lie in the set A _p . Further we show that when α is a root of an irreducible rational polynomial p(X) of degree ≠ 4 whose Galois group contains the full alternating group, α lies in A _p if and only if no real root of p(X) is positive.Received: 19 November 2004; revised: 9 February 2005     

The Galois group of the Eisenstein polynomial X 5 + aX + a

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     

A study on functional independence of the Iwasawa power series

We study a class of functional independences that the Iwasawa power series satisfy for both zero and non-zero characteristics. As results, we prove a generalization of Anglès and Ranieri [B. Anglès, G. Ranieri, On the linear independence of p$p$-adic L$L$-functions modulo p$p$, Ann. Inst. Fourier (Grenoble) 60 (5) (2010) 1831–1855] and transcendence of the Iwasawa power series over the rational functions for non-zero characteristics. We also verify that the power series is not a solution of any non-trivial linear differential equation with the coefficients of rational functions over p$p$-adic numbers.     

Class number 2 problem for certain real quadratic fields of Richaud-Degert type

In this paper we will apply Biró's method in [A. Biró, Yokoi's conjecture, Acta Arith. 106 (2003) 85-104; A. Biró, Chowla's conjecture, Acta Arith. 107 (2003) 179-194] to class number 2 problem of real quadratic fields of Richaud-Degert type and will show that there are exactly 4 real quadratic fields of the form with class number 2, where n²+1 is a even square free integer.     

A complete determination of Rabinowitsch polynomials

Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x²+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers x=x₀,x₀+1,…,x₀+s−1, |fm(x)| is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials fm(x).     

Maximal class numbers of CM number fields

Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis and Artin's conjecture on the entirety of Artin L-functions, we derive an upper bound (in terms of the discriminant) on the class number of any CM number field with maximal real subfield F. This bound is a refinement of a bound established by Duke in 2001. Under the same hypotheses, we go on to prove that there exist infinitely many CM-extensions of F whose class numbers essentially meet this improved bound and whose Galois groups are as large as possible.     

Additive Hilbert's Theorem 90 in the ring of algebraic integers

For a number field K, we give a complete characterization of algebraic numbers which can be expressed by a difference of two K-conjugate algebraic integers. These turn out to be the algebraic integers whose Galois group contains an element, acting as a cycle on some collection of conjugates which sum to zero. Hence there are no algebraic integers which can be written as a difference of two conjugate algebraic numbers but cannot be written as a difference of two conjugate algebraic integers. A generalization of the construction to a commutative ring is also given. Furthermore, we show that for n ?_ 3 there exist algebraic integers which can be written as a linear form in n K-conjugate algebraic numbers but cannot be written by the same linear form in K-conjugate algebraic integers.     

Square classes of totally positive units

In this article, we investigate some conditions for a real cyclic extension K over Q to satisfy the property that every totally positive unit of K is a square. As an application, we give a partial answer to Taussky's conjecture. We then extend our result to real abelian extensions of certain type.     

Noyaux de Tate et capitulation

Following Kahn, and Assim and Movahhedi, we look for bounds for the order of the capitulation kernels of higher K-groups of S-integers into abelian S-ramified p-extensions. The basic strategy is to change twists inside some Galois-cohomology groups, which is done via the comparison of Tate Kernels of higher order. We investigate two ways: a global one, valid for twists close to 0 (in a certain sense), and a local one, valid for twists close to 1 in cyclic extensions. The global method produces lower bounds for abelian p-extensions which are S-ramified, but not Zp-embeddable. The local method is close to that of [J. Assim, A. Movahhedi, Bounds for étale capitulation kernels, K-Theory 33 (2004) 199-213], but is improved to take into consideration what happens when S consists of only the p-places. In contrast to the first one, one can expect this second method to produce nontrivial lower bounds in certain Zp-extensions. For example, we construct Zp-extensions in which the capitulation kernel is as big as we want (when letting the twist vary). We also include a complete solution to the problem of comparing Tate Kernels.