Displaying similar documents to “On the Hilbert $2$-class field tower of some imaginary biquadratic number fields”

On the strongly ambiguous classes of some biquadratic number fields

Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous (2016)

Mathematica Bohemica

Similarity:

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

Similarity:

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...

Bicyclic commutator quotients with one non-elementary component

Daniel Mayer (2023)

Mathematica Bohemica

Similarity:

For any number field K with non-elementary 3 -class group Cl 3 ( K ) C 3 e × C 3 , e 2 , the punctured capitulation type ϰ ( K ) of K in its unramified cyclic cubic extensions L i , 1 i 4 , is an orbit under the action of S 3 × S 3 . By means of Artin’s reciprocity law, the arithmetical invariant ϰ ( K ) is translated to the punctured transfer kernel type ϰ ( G 2 ) of the automorphism group G 2 = Gal ( F 3 2 ( K ) / K ) of the second Hilbert 3 -class field of K . A classification of finite 3 -groups G with low order and bicyclic commutator quotient G / G ' C 3 e × C 3 , 2 e 6 , according to the algebraic...

Principalization algorithm via class group structure

Daniel C. Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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 i 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 = ( 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...

A note on the size Ramsey numbers for matchings versus cycles

Edy Tri Baskoro, Tomáš Vetrík (2021)

Mathematica Bohemica

Similarity:

For graphs G , F 1 , F 2 , we write G ( F 1 , F 2 ) if for every red-blue colouring of the edge set of G we have a red copy of F 1 or a blue copy of F 2 in G . The size Ramsey number r ^ ( F 1 , F 2 ) is the minimum number of edges of a graph G such that G ( F 1 , F 2 ) . Erdős and Faudree proved that for the cycle C n of length n and for t 2 matchings t K 2 , the size Ramsey number r ^ ( t K 2 , C n ) < n + ( 4 t + 3 ) n . We improve their upper bound for t = 2 and t = 3 by showing that r ^ ( 2 K 2 , C n ) n + 2 3 n + 9 for n 12 and r ^ ( 3 K 2 , C n ) < n + 6 n + 9 for n 25 .

The distribution of second p -class groups on coclass graphs

Daniel C. Mayer (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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 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 sums and products in a field

Guang-Liang Zhou, Zhi-Wei Sun (2022)

Czechoslovak Mathematical Journal

Similarity:

We study sums and products in a field. Let F be a field with ch ( F ) 2 , where ch ( F ) is the characteristic of F . For any integer k 4 , we show that any x F can be written as a 1 + + a k with a 1 , , a k F and a 1 a k = 1 , and that for any α F { 0 } we can write every x F as a 1 a k with a 1 , , a k F and a 1 + + a k = α . We also prove that for any x F and k { 2 , 3 , } there are a 1 , , a 2 k F such that a 1 + + a 2 k = x = a 1 a 2 k .

Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Maryam Khatami, Azam Babai (2018)

Czechoslovak Mathematical Journal

Similarity:

Let G be a finite group. An element g G is called a vanishing element if there exists an irreducible complex character χ of G such that χ ( g ) = 0 . Denote by Vo ( G ) the set of orders of vanishing elements of G . Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo ( G ) = Vo ( M ) and | G | = | M | . Then G M . We answer in affirmative this conjecture for M = S z ( q ) , where q = 2 2 n + 1 and either q - 1 , q - 2 q + 1 or q + 2 q + 1 is a prime number, and M = F 4 ( q ) , where...

On the unit group of a semisimple group algebra 𝔽 q S L ( 2 , 5 )

Rajendra K. Sharma, Gaurav Mittal (2022)

Mathematica Bohemica

Similarity:

We give the characterization of the unit group of 𝔽 q S L ( 2 , 5 ) , where 𝔽 q is a finite field with q = p k elements for prime p > 5 , and S L ( 2 , 5 ) denotes the special linear group of 2 × 2 matrices having determinant 1 over the cyclic group 5 .

More on exposed points and extremal points of convex sets in n and Hilbert space

Stoyu T. Barov (2023)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let 𝕍 be a separable real Hilbert space, k with k < dim 𝕍 , and let B be convex and closed in 𝕍 . Let 𝒫 be a collection of linear k -subspaces of 𝕍 . A point w B is called exposed by 𝒫 if there is a P 𝒫 so that ( w + P ) B = { w } . We show that, under some natural conditions, B can be reconstituted as the convex hull of the closure of all its exposed by 𝒫 points whenever 𝒫 is dense and G δ . In addition, we discuss the question when the set of exposed by some 𝒫 points forms a G δ -set.

On the 2 -class group of some number fields with large degree

Mohamed Mahmoud Chems-Eddin, Abdelmalek Azizi, Abdelkader Zekhnini (2021)

Archivum Mathematicum

Similarity:

Let d be an odd square-free integer, m 3 any integer and L m , d : = ( ζ 2 m , d ) . In this paper, we shall determine all the fields L m , d having an odd class number. Furthermore, using the cyclotomic 2 -extensions of some number fields, we compute the rank of the 2 -class group of L m , d whenever the prime divisors of d are congruent to 3 or 5 ( mod 8 ) .

The potential-Ramsey number of K n and K t - k

Jin-Zhi Du, Jian Hua Yin (2022)

Czechoslovak Mathematical Journal

Similarity:

A nonincreasing sequence π = ( d 1 , ... , d n ) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of π . Given two graphs G 1 and G 2 , A. Busch et al. (2014) introduced the potential-Ramsey number of G 1 and G 2 , denoted by r pot ( G 1 , G 2 ) , as the smallest nonnegative integer m such that for every m -term graphic sequence π , there is a realization G of π with G 1 G or with G 2 G ¯ , where G ¯ is the complement of G . For t 2 and 0 k t 2 , let K t - k be the graph...