On a Problem of Hasse for Certain Imaginary Abelian Fields (original) (raw)

Monogenesis of the rings of integers in certain imaginary abelian fields

Nagoya Mathematical Journal, 2002

In this paper we consider a subfield K in a cyclotomic field km of conductor m such that [km: K] = 2 in the cases of m = lpn with a prime p, where l = 4 or p > l = 3. Then the theme is to know whether the ring of integers in K has a power basis or does not.

On Monogeneity of Cyclic Quartic Fields of Prime Conductor

In this paper we want to give a new proof for non-monogenesis of any cyclic quartic fields K over the rationals Q of prime conductor congruent 1 modulo 4 except for the 5th cyclotomic field Q(exp(2 i/5)). π This phenomenon was once proved using the Gauss sum attached to a quartic character by the second author. For Hasse's problem to determine whether an algebraic number field whose ring of integers has a power integral basis or not, Y. Motoda and the second author proved that infinitely many 2-elementary abelian fields with degree 8 have no power integral basis by solving Diophantine equations associated to seven quadratic subfields of an octic field L except for the 24th cyclotomic field Q(exp(2 i/24)). π Our emphasis is applying a single simultaneous linear Diophantine equation to give a totally different and most succinct proof rather than the previous one for non-monogenesis of cyclic quartic fields K with prime conductor.

On a Problem of Hasse (Algebraic Number Theory and Related Topics 2007)

RIMS Kokyuroku …, 2009

In this article we shall construct a new family of cyclic quartic fields K with odd composite conductors, which give an affirmative solution to a Problem of Hasse(Problem 6 in [12, p. 529]); indeed our family consists of cyclic quartic fields whose ring Z K of integers are generated by a single element ξ over Z. We will find an integer ξ in K by the two different ways; one of which is based on an integral basis of Z K and the other is done on a field basis of K.

Some results on a special type of real quadratic fields

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS, 2019

Some results on a special type of real quadratic fields In this paper, we determine the real quadratic fields Q(√ d) coincide with positive square-free integers d including the continued fraction expansion form of w d =    a0; 7, 7,. .. , 7 −1 , a   . Furthermore, we deal with determining fundamental units and Yokoi's d-invariants n d and m d in the relation to continued fraction expansion of w d where (d) is a period length of w d for the such type of real quadratic number fields Q(√ d). The present paper improve the theory of fundamental unit which generates the unit group of real quadratic fields and also determine the special form of continued fraction expansion of integral basis element in real quadratic fields.

On biquadratic fields that admit unit power integral basis

Acta Mathematica Hungarica, 2011

In this paper we consider bi-quadratic number fields, whose maximal orders have power integral bases consisting of units. We prove an effective and efficient criteria to decide whether the maximal order of a bi-quadratic field has a unit power integral bases or not. In particular we can determine all trivial fields, whose maximal orders have a unit power integral bases.

Capitulation in the Absolutely Abelian Extensions of some Number Fields II

Acta Mathematica Vietnamica, 2016

We study the capitulation of 2-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields k = Q(√ pq1q2, i), where i = √ −1 and q1 ≡ q2 ≡ −p ≡ −1 (mod 4) are different primes. For each of the three quadratic extensions K/k inside the absolute genus field k (*) of k, we compute the capitulation kernel of K/k. Then we deduce that each strongly ambiguous class of k/Q(i) capitulates already in k (*) .

On fields with the Property (B)

HAL (Le Centre pour la Communication Scientifique Directe), 2014

Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let us assume that G/Z(G) has finite exponent. We show that L has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on L * (outside the set of roots of unity) is bounded from below by an absolute constant. We discuss some feature of Property (B): stability by algebraic extensions, relations with field arithmetic. As a as a side result, we prove that the Galois group over Q of the compositum of all totally real fields is torsion free.