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     

Real Quadratic Fields with Class Number Divisible by 5 or 7

We shall show that the number of real quadratic fields whose absolute discriminant is ≤ x and whose class number is divisible by 5 or 7 is improving the existing best known bound for g = 5 and for g = 7 of Yu (J Number Theory 97:35–44, 2002).This work was supported by KRF-R08-2003-000-10243-0 and partially by KRF-2005-070-C00004.     

The Index of a Cyclic Quartic Field

Let K be a cyclic quartic field. Let i(K) denote the index of K. It is known that i(K){1, 2, 3, 4, 6, 12}. In Part 1 of this paper we show that i(K) assumes all of these values and we give necessary and sufficient conditions for each to occur. In Part 2 an asymptotic formula is given for the number of cyclic quartic fields with discriminant x and i(K)=i for each i{1, 2, 3, 4, 6, 12}.Received May 6, 2002; in revised form December 24, 2002 Published online June 23, 2003     

On formulas for the index of the circular distributions

A circular distribution is a Galois equivariant map ψ from the roots of unity μ _∞ to an algebraic closure of such that ψ satisfies product conditions, for ϵ ∈ μ _∞ and , and congruence conditions for each prime number l and with (l, s) = 1, modulo primes over l for all , where μ _l and μ _s denote respectively the sets of lth and sth roots of unity. For such ψ, let be the group generated over by and let be , where U _s denotes the global units of . We give formulas for the indices and of and inside the circular numbers P _s and units C _s of Sinnott over . This work was supported by the SRC Program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R11-2007-035-01001-0). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00455).     

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.     

Quadratic equations over finite fields and class numbers of real quadratic fields

Letp be an odd prime and the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field (wherep1 (mod 4)) over the field of rational numbers can be expressed by means of these formulas.     

A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

The number of steps in the euclidean algorithm over complex quadratic fields

We obtain upper and lower bounds for the number of divisions in the Euclidean algorithm, for almost all pairs of algebraic integers lying in the complex quadratic fields (–m), form=1, 2, 3, 7 and 11. In addition, the order of the average length for almost all such pairs is deduced.     

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     

Mahler measures in a field are dense modulo 1

Let K be a number field. We prove that the set of Mahler measures M(α), where α runs over every element of K, modulo 1 is everywhere dense in [0, 1], except when or , where D is a positive integer. In the proof, we use a certain sequence of shifted Pisot numbers (or complex Pisot numbers) in K and show that the corresponding sequence of their Mahler measures modulo 1 is uniformly distributed in [0, 1]. Received: 24 March 2006     

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 .     

Caliber number of real quadratic fields

We obtain lower bound of caliber number of real quadratic field using splitting primes in K. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if d is not 5 modulo 8. In both cases, we don't rely on the assumption on ζK(1/2).     

Dual vectors and lower bounds for the nearest lattice point problem

We prove that given a point outside a given latticeL then there is a dual vector which gives a fairly good estimate for how far from the lattice the vector is. To be more precise, there is a set of translated hyperplanesH _i, such thatL _iH_i andd( _iH_i)(6n ²+1)^–1 d( ,L).Supported by an IBM fellowship.     

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.     

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.     

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.     

Capitulation of the 2‐class group of some cyclic number fields with large degree

We study the capitulation problem of the 2‐class group of some cyclic number fields M with large degree and 2‐class group isomorphic to . Precisely, we give the structure of the Galois group of the maximal unramified 2‐extension over M.     

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.     

Construction of a certain circular unit and its applications

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.