Integer valued polynomials over a number field (original) (raw)

A remark on class numbers of number field extensions

Proceedings of the American Mathematical Society, 1969

In this note we study some cases in which the structure of the Galois group of an extension of number fields gives information about the relation between the ideal class groups of these fields. The letters K, L, etc. will denote finite extension fields of the rationals and the letter p will denote a rational prime. If A is an abelian group we let its order be denoted by |^4| and its p-sylow subgroup by Ap. The class group of L will be written Cl¿.

On Arithmetically Equivalent Number Fields of Small Degree

Lecture Notes in Computer Science, 2002

For each integer n, let Sn be the set of all class number quotients h(K)/h(K ) for number fields K and K of degree n with the same zeta-function. In this note we will give some explicit results on the finite sets Sn, for small n. For example, for every x ∈ Sn with n ≤ 15, x or x −1 is an integer that is a prime power dividing 2 14 · 3 6 · 5 3 .

On the construction of class fields

The Open Book Series, 2019

Class field theory is an important tool in number theory. We discuss improvements to the computation of ray class groups, congruence subgroups and class fields, which are fundamental building blocks of constructive class field theory. As an application and to illustrate the power of our new techniques, we find new fields with minimal discriminant having prescribed Galois group and signature.

On the indices in number fields and their computation for small degrees

Applicable Analysis and Discrete Mathematics, 2021

Let K be a number field. We investigate the indices I(K) and i(K) of K introduced respectively by Dedekind and Gunji-McQuillan. Let n be a positif integer, we then prove that for any prime p ? n, there exists K a number field of degree n over Q such that p divide i(K). This result is an analogue to Bauer''s one for i(K). We compute I(K) and i(K) for cubic fields and infinite families of simplest number fields of degree less than 7. We solve questions and disprove the conjecture stated in [1].