On $p^2$-Ranks in the Class Field Tower Problem

Christian Maire; Cam McLeman

Displaying similar documents to “On $p^{2}$ -Ranks in the Class Field Tower Problem”

Subfields of henselian valued fields

Ramneek Khassa, Sudesh K. Khanduja (2010)

Colloquium Mathematicae

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Let (K,v) be a henselian valued field of arbitrary rank which is not separably closed. Let k be a subfield of K of finite codimension and $v_{k}$ be the valuation obtained by restricting v to k. We give some necessary and sufficient conditions for $(k, v_{k})$ to be henselian. In particular, it is shown that if k is dense in its henselization, then $(k, v_{k})$ is henselian. We deduce some well known results proved in this direction through other considerations.

On the structure of the Galois group of the Abelian closure of a number field

Georges Gras (2014)

Journal de Théorie des Nombres de Bordeaux

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From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_{K}$ , we study any such issue for arbitrary number fields $K$ . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$ -adic obstructions coming from the global units of $K$ . By restriction to the $p$ -Sylow subgroups of $A_{K}$ and assuming the Leopoldt conjecture we...

Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

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The relative class number $H_{d} (f)$ of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of $_{f}$ and $_{K}$ , where $_{K}$ denotes the ring of integers of K and $_{f}$ is the order of conductor f given by $ℤ + f_{K}$ . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the...

A $C^{*}$ -Algebra Approach to Field Theory

D. Kastler (1968)

Recherche Coopérative sur Programme n°25

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

Daniel C. Mayer (2013)

Journal de Théorie des Nombres de Bordeaux

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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...

On the strongly ambiguous classes of some biquadratic number fields

Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous (2016)

Mathematica Bohemica

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We study the capitulation of $2$ -ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $𝕜 = ℚ (\sqrt{2 p q}, i)$ , where $i = \sqrt{- 1}$ and $p \equiv - q \equiv 1 (mod 4)$ are different primes. For each of the three quadratic extensions $𝕂 / 𝕜$ inside the absolute genus field $𝕜^{(*)}$ of $𝕜$ , we determine a fundamental system of units and then compute the capitulation kernel of $𝕂 / 𝕜$ . The generators of the groups ${Am}_{s} (𝕜 / F)$ and $Am (𝕜 / F)$ are also determined from which we deduce that $𝕜^{(*)}$ is smaller than the relative genus field ${(𝕜 / ℚ (i))}^{*}$ . Then we prove...

Governing fields for the 2-classgroup of $Q (s q r t - q_{1} q_{2} p)$ and a related reciprocity law

Patrick Morton (1990)

Acta Arithmetica

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Principalization algorithm via class group structure

Daniel C. Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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For an algebraic number field $K$ with $3$ -class group ${Cl}_{3} (K)$ of type $(3, 3)$ , the structure of the $3$ -class groups ${Cl}_{3} (N_{i})$ of the four unramified cyclic cubic extension fields $N_{i}$ , $1 \leq i \leq 4$ , of $K$ is calculated with the aid of presentations for the metabelian Galois group $G_{3}^{2} (K) = Gal (F_{3}^{2} (K) | K)$ of the second Hilbert $3$ -class field $F_{3}^{2} (K)$ of $K$ . In the case of a quadratic base field $K = ℚ (\sqrt{D})$ it is shown that the structure of the $3$ -class groups of the four $S_{3}$ -fields $N_{1}, ..., N_{4}$ frequently determines the type of principalization of the $3$ -class group of $K$ in $N_{1}, ..., N_{4}$ . This...

Small discriminants of complex multiplication fields of elliptic curves over finite fields

Igor E. Shparlinski (2015)

Czechoslovak Mathematical Journal

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We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves $E$ over a prime finite field $𝔽_{p}$ of $p$ elements, such that the discriminant $D (E)$ of the quadratic number field containing the endomorphism ring of $E$ over $𝔽_{p}$ is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).

Isomorphisms of algebraic number fields

Mark van Hoeij, Vivek Pal (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $ℚ (α)$ and $ℚ (β)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $ℚ (β) \to ℚ (α)$ . The algorithm is particularly efficient if there is only one isomorphism.

$H^{p}$ -spaces of conjugate systems on local fields

Jia Chao (1974)

Studia Mathematica

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Kronecker’s solution of Pell’s equation for CM fields

Riad Masri (2013)

Annales de l’institut Fourier

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We generalize Kronecker’s solution of Pell’s equation to CM fields $K$ whose Galois group over $ℚ$ is an elementary abelian 2-group. This is an identity which relates CM values of a certain Hilbert modular function to products of logarithms of fundamental units. When $K$ is imaginary quadratic, these CM values are algebraic numbers related to elliptic units in the Hilbert class field of $K$ . Assuming Schanuel’s conjecture, we show that when $K$ has degree greater than 2 over $ℚ$ these CM values...

Displaying similar documents to “On p 2 -Ranks in the Class Field Tower Problem”

Displaying similar documents to “On $p^{2}$ -Ranks in the Class Field Tower Problem”