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Displaying similar documents to “On the compositum of all degree d extensions of a number field”

When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Stéphane R. Louboutin (2020)

Czechoslovak Mathematical Journal

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Let ε be an algebraic unit of the degree n 3 . Assume that the extension ( ε ) / is Galois. We would like to determine when the order [ ε ] of ( ε ) is Gal ( ( ε ) / ) -invariant, i.e. when the n complex conjugates ε 1 , , ε n of ε are in [ ε ] , which amounts to asking that [ ε 1 , , ε n ] = [ ε ] , i.e., that these two orders of ( ε ) have the same discriminant. This problem has been solved only for n = 3 by using an explicit formula for the discriminant of the order [ ε 1 , ε 2 , ε 3 ] . However, there is no known similar formula for n > 3 . In the present paper, we put forward and...

Bicyclic commutator quotients with one non-elementary component

Daniel Mayer (2023)

Mathematica Bohemica

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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...

Variations on a question concerning the degrees of divisors of x n - 1

Lola Thompson (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we examine a natural question concerning the divisors of the polynomial x n - 1 : “How often does x n - 1 have a divisor of every degree between 1 and n ?” In a previous paper, we considered the situation when x n - 1 is factored in [ x ] . In this paper, we replace [ x ] with 𝔽 p [ x ] , where p is an arbitrary-but-fixed prime. We also consider those n where this condition holds for all p .

Piecewise hereditary algebras under field extensions

Jie Li (2021)

Czechoslovak Mathematical Journal

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Let A be a finite-dimensional k -algebra and K / k be a finite separable field extension. We prove that A is derived equivalent to a hereditary algebra if and only if so is A k K .

On sums and products in a field

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

Czechoslovak Mathematical Journal

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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 .

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi (2017)

Czechoslovak Mathematical Journal

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Let G be a group and ω ( G ) be the set of element orders of G . Let k ω ( G ) and m k ( G ) be the number of elements of order k in G . Let nse ( G ) = { m k ( G ) : k ω ( G ) } . Assume r is a prime number and let G be a group such that nse ( G ) = nse ( S r ) , where S r is the symmetric group of degree r . In this paper we prove that G S r , if r divides the order of G and r 2 does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...

Realizable Galois module classes over the group ring for non abelian extensions

Nigel P. Byott, Bouchaïb Sodaïgui (2013)

Annales de l’institut Fourier

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Given an algebraic number field k and a finite group Γ , we write ( O k [ Γ ] ) for the subset of the locally free classgroup Cl ( O k [ Γ ] ) consisting of the classes of rings of integers O N in tame Galois extensions N / k with Gal ( N / k ) Γ . We determine ( O k [ Γ ] ) , and show it is a subgroup of Cl ( O k [ Γ ] ) by means of a description using a Stickelberger ideal and properties of some cyclic codes, when k contains a root of unity of prime order p and Γ = V C , where V is an elementary abelian group of order p r and C is a cyclic group of order m &gt; 1 acting faithfully...