A density result for some imaginary quadratic fields with infinite Hilbert 2-class field towers

This paper shows that a positive proportion of the imaginary quadratic fields with 2-class rank equal to 3 have 4-class rank equal to zero and infinite Hilbert 2-class field towers. Received: 14 January 2003     

On the 2-class field tower of some imaginary biquadratic number fields

Let be an imaginary biquadratic number field with Cl_k,2, the 2-class group of k, isomorphic to Z/2Z × Z/2^mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1. 2000 Mathematics Subject Classification Primary—11R29     

On the Hilbert 2-class field tower of some abelian 2-extensions over the field of rational numbers

It is well known by results of Golod and Shafarevich that the Hilbert 2-class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian 2-extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian 2-extension over ? in which eight primes ramify and one of theses primes ≡ ?1 (mod 4), the Hilbert 2-class field tower is infinite.     

On 2-ciass field towers of some imaginary quadratic number fields

We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2, 2, 2) whose Hilbert 2-class fields are finite.     

Unit groups of some multiquadratic number fields and 2-class groups

Periodica Mathematica Hungarica - Let $$pequiv -q equiv 5pmod 8$$ be two prime integers. In this paper, we investigate the unit groups of the fields $$ L_1 =mathbb {Q}(sqrt{2}, sqrt{p},...     

On the Galois groups of the 2-class towers of some imaginary quadratic fields

Let and knr,2 be the maximal unramified 2-extension of k. To show that knr,2/k is finite, Michael Bush gave 8 possible presentations of finite groups for G=Gal(knr,2/k). However, his methods did not further isolate G. We eliminate 4 of the possibilities, and explain how to isolate G, although carrying out the latter strategy is beyond current technological capabilities. We also discuss related examples.     

Computation of Galois groups associated to the 2-class towers of some quadratic fields

The p-group generation algorithm from computational group theory is used to obtain information about large quotients of the pro-2 group for with d=−445,−1015,−1595,−2379. In each case we are able to narrow the identity of G down to one of a finite number of explicitly given finite groups. From this follow several results regarding the corresponding 2-class tower.     

Imaginary quadratic fields of class number 2

A conjecture concerning linear forms in the logarithms of algebraic numbers is made. It is shown that this conjecture allows an effective determination of all imaginary quadratic fields of class number 2.     

Densities for 4-class ranks of quadratic function fields

Let F=Fq(T) be a rational function field of odd characteristic, and fix a positive integer t. In this article we study the family of quadratic function fields , where D is a polynomial over Fq of odd degree having t distinct irreducible factors. The 4-class rank r₄(K) is the rank of the 4-torsion of the group of divisor classes of K, and it is known that 0?r₄(K)?t−1. For fixed r we compute the proportion of such fields K satisfying r₄(K)=r, and in particular we determine the behaviour of this value as t→∞. We will need some asymptotic results for these computations, in particular the number of polynomials D as above whose irreducible factors fulfill certain parity and quadratic residue conditions.     

Asymptotic behavior of number fields with prescribed ℓ-class numbers

Let K be a cyclic Galois extension of the rational numbers Q of degree ?, where ? is a prime number. Let h_? denote the order of the Sylow ?-subgroup of the ideal class group of K. If h_? = ?^s(s ≥ 0), it is known that the number of (finite) primes that ramify in K/Q is at most s + 1 (or s + 2 if K is real quadratic). This paper shows that “most” of these fields K with h_? = ?^s have exactly s + 1 ramified primes (or s + 2 ramified primes if K is real quadratic). Furthermore the Sylow ?-subgroup of the ideal class group is elementary abelian when h_? = ?^s and there are s + 1 ramified primes (or s + 2 ramified primes if K is real quadratic).     

Abelian 2-class field towers over the cyclotomic {\mathbb {Z}_2}-extensions of imaginary quadratic fields

For the cyclotomic \mathbb Z₂{\mathbb Z_2}-extension k _∞ of an imaginary quadratic field k, we consider whether the Galois group G(k _∞) of the maximal unramified pro-2-extension over k _∞ is abelian or not. The group G(k _∞) is abelian if and only if the nth layer of the \mathbb Z₂{\mathbb {Z}_2}-extension has abelian 2-class field tower for all n ≥ 1. The purpose of this paper is to classify all such imaginary quadratic fields k in part by using Iwasawa polynomials.     

Class number one quadratic fields and solvability of some Pellian equations

Consider these two types of positive square-free integers d≠ 1 for which the class number h of the quadratic field Q(√d) is odd: (1) d is prime∈ 1(mod 8), or d=2q where q is prime ≡ 3 (mod 4), or d=qr where q and r are primes such that q≡ 3 (mod 8) and r≡ 7 (mod 8); (2) d is prime ≡ 1 (mod 8), or d=qr where q and r are primes such that q≡r≡ 3 or 7 (mod 8). For d of type (2) (resp. (1)), let Π be the set of all primes (resp. odd primes) p∈N satisfying (d/p) = 1. Also, let δ :=0 (resp. δ :=1) if d≡ 2,3 (mod 4) (resp. d≡ 1 (mod 4)). Then the following are equivalent: (a) h=1; (b) For every p∈П at least one of the two Pellian equations Z ²-dY ² = ±4^δ p is solvable in integers. (c) For every p∈П the Pellian equation W ²-dV ² = 4^δ p ² has a solution (w,v) in integers such that gcd (w,v) divides 2^δ.     

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).