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Displaying 1 – 20 of 38

Tate sequences and lower bounds for ranks of class groups

Cornelius Greither (2013)

Acta Arithmetica

Tate sequences play a major role in modern algebraic number theory. The extension class of a Tate sequence is a very subtle invariant which comes from class field theory and is hard to grasp. In this short paper we demonstrate that one can extract information from a Tate sequence without knowing the extension class in two particular situations. For certain totally real fields K we will find lower bounds for the rank of the ℓ-part of the class group Cl(K), and for certain CM fields we will find lower...

The 4-class ranks of quadratic fields related to the Weyl group algebra.

Frank III Gerth (1984)

Inventiones mathematicae

The class number one problem for some non-abelian normal CM-fields of degree $24$

F. Lemmermeyer, S. Louboutin, R. Okazaki (1999)

Journal de théorie des nombres de Bordeaux

We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to $𝒜_{4}$ , the alternating group of degree $4$ and order $12$ . There are two such fields with Galois group $𝒜_{4} \times 𝒞_{2}$ (see Theorem 14) and at most one with Galois group SL $_{2} (𝔽_{3})$ (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number $1$ .

The class number one problem for the dihedral and dicyclic CM-fields

Stéphane Louboutin (1999)

Colloquium Mathematicae

We recall the determination of all the dihedral CM-fields with relative class number one, and prove that dicyclic CM-fields have relative class numbers greater than one.

The class number one problem for the non-abelian normal CM-fields of degree 16

Stéphane Louboutin (1997)

Acta Arithmetica

The class number one problem for the non-abelian normal CM-fields of degree 24 and 40

Young-Ho Park (2002)

Acta Arithmetica

The class number one problem for the non-normal sextic CM-fields. Part 2

Gérard Boutteaux, Stéphane Louboutin (2002)

Acta Mathematica et Informatica Universitatis Ostraviensis

The class number one problem for the real quadratic fields ℚ(√((an)²+4a))

András Biró, Kostadinka Lapkova (2016)

Acta Arithmetica

We solve unconditionally the class number one problem for the 2-parameter family of real quadratic fields ℚ(√d) with square-free discriminant d = (an)²+4a for positive odd integers a and n.

The Cohen-Lenstra heuristics, moments and $p^{j}$ -ranks of some groups

Christophe Delaunay, Frédéric Jouhet (2014)

Acta Arithmetica

This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to É. Fouvry and J. Klüners which proves that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture for the probability laws of $p^{j}$ -ranks of Selmer groups...

The complete determination of wide Richaud-Degert types which are not 5 modulo 8 with class number one

Jungyun Lee (2009)

Acta Arithmetica

The digits of 1/p in connection with class number factors

Kurt Girstmair (1994)

Acta Arithmetica

The Diophantine equation $x^{2} + q^{m} = p^{n}$

Nobuhiro Terai (1993)

Acta Arithmetica

The distribution of second $p$ -class groups on coclass graphs

Daniel C. Mayer (2013)

Journal de Théorie des Nombres de Bordeaux

General concepts and strategies are developed for identifying the isomorphism type of the second $p$ -class group $G = Gal (F_{p}^{2} (K) | K)$ , that is the Galois group of the second Hilbert $p$ -class field $F_{p}^{2} (K)$ , of a number field $K$ , for a prime $p$ . The isomorphism type determines the position of $G$ on one of the coclass graphs $𝒢 (p, r)$ , $r \geq 0$ , in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$ -class group ${Cl}_{p} (K)$ , the position of $G$ is restricted to certain admissible branches of coclass...

The field descent and class groups of CM-fields

Bernhard Schmidt (2005)

Acta Arithmetica

The Heegner-Stark-Baker-Deuring-Siegel Theorem.

S. Chowla (1970)

Journal für die reine und angewandte Mathematik

The homology of cyclic branched covers of S3.

James F. Davis (1995)

Mathematische Annalen

The homology of irregular dihedral branched covers of S3.

James F. Davis (1995)

Mathematische Annalen

The imaginary abelian number fields with class numbers equal to their genus class numbers

Ku-Young Chang, Soun-Hi Kwon (2000)

Journal de théorie des nombres de Bordeaux

We know that there exist only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Such non-quadratic cyclic number fields are completely determined in [Lou2,4] and [CK]. In this paper we determine all non-cyclic abelian number fields with class numbers equal to their genus class numbers, thus the one class in each genus problem is solved, except for the imaginary quadratic number fields.

The imaginary cyclic sextic fields with class numbers equal to their genus class numbers

Stéphane Louboutin (1998)

Colloquium Mathematicae

It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic sextic fields with class numbers equal to their genus class numbers.

The imaginary quadratic fields of class number 4

Steven Arno (1992)

Acta Arithmetica

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