Brauer groups of genus zero extensions of number fields

We determine the isomorphism class of the Brauer groups of certain nonrational genus zero extensions of number fields. In particular, for all genus zero extensions of the rational numbers that are split by , .

     

Remark on infinite unramified extensions of number fields with class number one

We modify an idea of Maire to construct biquadratic number fields with small root discriminants, class number one, and having an infinite, necessarily non-solvable, strictly unramified Galois extension.     

On the number of certain Galois extensions of local fields

In this paper, we will calculate the number of Galois extensions of local fields with Galois group or .

     

Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ^-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq. Work supported in part by an operating grant from NSERC (Canada).     

Steinitz classes of tamely ramified Galois extensions of algebraic number fields

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers Ok of k, which, together with the degree [K:k] of the extension determines the Ok-module structure of OK. We call Rt(k,G) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A^′-groups inductively, starting with abelian groups and then considering semidirect products of A^′-groups with abelian groups of relatively prime order and direct products of two A^′-groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A^′-groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine Rt(k,Dn) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).     

A note on abundant central extensions of number fields and Scholz's solutions of number knots

Archiv der Mathematik -     

On 3-class groups of cyclic cubic extensions of certain number fields

Let F be a (finite) algebraic number field, and let K be a cyclic cubic extension of F. Assuming that the 3-class group of F is trivial, we determine the rank of the 3-class group S_K of K, and we obtain some additional information about the structure of S_K.     

Definability of the ring of integers in pro-p galois extensions of number fields

For some infinite algebraic extensionsM of the field of rational numbers we show that the integral closure of ℤ is first-order definable. As an application we prove that the Pythagorean hull of ℚ is an undecidable field.     

Cyclotomic splitting fields and strict cohomological dimension

The purpose of this note is to characterize fields of strict cohomological dimension two in terms of Brauer groups.     

Normal integral bases of ∞ -ramified abelian extensions of totally real number fields

LetF be a totally real number field and [ a product of some real primes ofF. J. Brinkhuis gave a necessary condition for a finite abelian extension ofF which is unramified outside [ to have a normal integral basis. We consider the converse of his result and give a necessary and sufficient condition. Furthermore, we concretely express it whenF is a real quadratic field or a cyclic cubic field.     

On Galois structure of the integers in elementary abelian extensions of local number fields

Let p be an odd prime number and k a finite extension of Qp. Let K/k be a totally ramified elementary abelian Kummer extension of degree p² with Galois group G. We determine the isomorphism class of the ring of integers in K as an oG-module under some assumptions. The obtained results imply there exist extensions whose rings are ZpG-isomorphic but not oG-isomorphic, where Zp is the ring of p-adic integers. Moreover we obtain conditions that the rings of integers are free over the associated orders and give extensions whose rings are not free.     

Cyclotomic matrices and hypergeometric functions over finite fields

With the help of hypergeometric functions over finite fields, we study some arithmetic properties of cyclotomic matrices involving characters and binary quadratic forms over finite fields. Also, we confirm some related conjectures posed by Zhi-Wei Sun.     

G-primitive extensions of differential fields

A classification is given of the G-primitive extensions of a partial differential field of characteristic zero with an algebraically closed field of constants and a connected algebraic group G.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 425–434, March, 1978.