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Displaying similar documents to “Exponent of class group of certain imaginary quadratic fields”

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 + 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...

Positivity of quadratic base change L -functions

Hervé Jacquet, Chen Nan (2001)

Bulletin de la Société Mathématique de France

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We show that certain quadratic base change L -functions for Gl ( 2 ) are non-negative at their center of symmetry.

Real quadratic number fields with metacyclic Hilbert 2 -class field tower

Said Essahel, Ahmed Dakkak, Ali Mouhib (2019)

Mathematica Bohemica

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We begin by giving a criterion for a number field K with 2-class group of rank 2 to have a metacyclic Hilbert 2-class field tower, and then we will determine all real quadratic number fields ( d ) that have a metacyclic nonabelian Hilbert 2 -class field tower.

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 = ( 2 p q , i ) , where i = - 1 and p - q 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 / ( i ) ) * . Then we prove...

Lower bound for class numbers of certain real quadratic fields

Mohit Mishra (2023)

Czechoslovak Mathematical Journal

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Let d be a square-free positive integer and h ( d ) be the class number of the real quadratic field ( d ) . We give an explicit lower bound for h ( n 2 + r ) , where r = 1 , 4 . Ankeny and Chowla proved that if g > 1 is a natural number and d = n 2 g + 1 is a square-free integer, then g h ( d ) whenever n > 4 . Applying our lower bounds, we show that there does not exist any natural number n > 1 such that h ( n 2 g + 1 ) = g . We also obtain a similar result for the family ( n 2 g + 4 ) . As another application, we deduce some criteria for a class group of prime power order to be...

Weighted Erdős-Kac type theorem over quadratic field in short intervals

Xiaoli Liu, Zhishan Yang (2022)

Czechoslovak Mathematical Journal

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Let 𝕂 be a quadratic field over the rational field and a 𝕂 ( n ) be the number of nonzero integral ideals with norm n . We establish Erdős-Kac type theorems weighted by a 𝕂 ( n ) l and a 𝕂 ( n 2 ) l of quadratic field in short intervals with l + . We also get asymptotic formulae for the average behavior of a 𝕂 ( n ) l and a 𝕂 ( n 2 ) l in short intervals.

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...

Eigenspaces of the ideal class group

Cornelius Greither, Radan Kučera (2014)

Annales de l’institut Fourier

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The aim of this paper is to prove an analog of Gras’ conjecture for an abelian field F and an odd prime p dividing the degree [ F : ] assuming that the p -part of Gal ( F / ) group is cyclic.

Capturing forms in dense subsets of finite fields

Brandon Hanson (2013)

Acta Arithmetica

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An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field q . Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size...