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Displaying similar documents to “On some equations over finite fields”

Weber's class invariants revisited

Reinhard Schertz (2002)

Journal de théorie des nombres de Bordeaux

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Let K be a quadratic imaginary number field of discriminant d . For t let 𝔒 t denote the order of conductor t in K and j ( 𝔒 t ) its modular invariant which is known to generate the ring class field modulo t over K . The coefficients of the minimal equation of j ( 𝔒 t ) being quite large Weber considered in [We] the functions f , f 1 , f 2 , γ 2 , γ 3 defined below and thereby obtained simpler generators of the ring class fields. Later on the singular values of these functions played a crucial role in Heegner’s solution [He] of...

On units of some fields of the form ( 2 , p , q , - l )

Mohamed Mahmoud Chems-Eddin (2023)

Mathematica Bohemica

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Let p 1 ( mod 8 ) and q 3 ( mod 8 ) be two prime integers and let { - 1 , p , q } be a positive odd square-free integer. Assuming that the fundamental unit of ( 2 p ) has a negative norm, we investigate the unit group of the fields ( 2 , p , q , - ) .

Note on the Hilbert 2-class field tower

Abdelmalek Azizi, Mohamed Mahmoud Chems-Eddin, Abdelkader Zekhnini (2022)

Mathematica Bohemica

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Let k be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields 𝕜 = ( d , - 1 ) , which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).

Extension of CR functions to «wedge type» domains

Andrea D'Agnolo, Piero D'Ancona, Giuseppe Zampieri (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let X be a complex manifold, S a generic submanifold of X R , the real underlying manifold to X . Let Ω be an open subset of S with Ω analytic, Y a complexification of S . We first recall the notion of Ω -tuboid of X and of Y and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ¯ b to the extendability of C R functions on Ω to Ω -tuboids of X . Next, if X has complex dimension 2, we give results...

The unit group of some fields of the form ( 2 , p , q , - l )

Moha Ben Taleb El Hamam (2024)

Mathematica Bohemica

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Let p and q be two different prime integers such that p q 3 ( mod 8 ) with ( p / q ) = 1 , and l a positive odd square-free integer relatively prime to p and q . In this paper we investigate the unit groups of number fields 𝕃 = ( 2 , p , q , - l ) .

Legendre polynomials and supercongruences

Zhi-Hong Sun (2013)

Acta Arithmetica

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Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), P [ p / 6 ] ( t ) - ( 3 / p ) x = 0 p - 1 ( ( x ³ - 3 x + 2 t ) / p ) ( m o d p ) and ( x = 0 p - 1 ( ( x ³ + m x + n ) / p ) ) ² ( ( - 3 m ) / p ) k = 0 [ p / 6 ] 2 k k 3 k k 6 k 3 k ( ( 4 m ³ + 27 n ² ) / ( 12 ³ · 4 m ³ ) ) k ( m o d p ) , where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning k = 0 p - 1 2 k k 3 k k 6 k 3 k / m k ( m o d p ² ) , where m is an integer not divisible by p.

Inequalities for the arithmetical functions of Euler and Dedekind

Horst Alzer, Man Kam Kwong (2020)

Czechoslovak Mathematical Journal

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For positive integers n , Euler’s phi function and Dedekind’s psi function are given by φ ( n ) = n p n p prime 1 - 1 p and ψ ( n ) = n p n p prime 1 + 1 p , respectively. We prove that for all n 2 we have 1 - 1 n n - 1 1 + 1 n n + 1 φ ( n ) n φ ( n ) ψ ( n ) n ψ ( n ) and φ ( n ) n ψ ( n ) ψ ( n ) n φ ( n ) 1 - 1 n n + 1 1 + 1 n n - 1 . The sign of equality holds if and only if n is a prime. The first inequality refines results due to Atanassov (2011) and Kannan & Srikanth (2013).

Norm inequalities for the difference between weighted and integral means of operator differentiable functions

Silvestru Sever Dragomir (2020)

Archivum Mathematicum

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Let f be a continuous function on I and A , B 𝒮𝒜 I H , the convex set of selfadjoint operators with spectra in I . If A B and f , as an operator function, is Gateaux differentiable on [ A , B ] : = ( 1 - t ) A + t B t 0 , 1 , while p : 0 , 1 is Lebesgue integrable, then we have the inequalities 0 1 p τ f 1 - τ A + τ B d τ - 0 1 p τ d τ 0 1 f 1 - τ A + τ B d τ 0 1 τ ( 1 - τ ) | τ 1 p s d s 1 - τ - 0 τ p s d s τ | f 1 - τ A + τ B B - A d τ 1 4 0 1 | τ 1 p s d s 1 - τ - 0 τ p s d s τ | f 1 - τ A + τ B B - A d τ , where f is the Gateaux derivative of f .

The conductor of a cyclic quartic field using Gauss sums

Blair K. Spearman, Kenneth S. Williams (1997)

Czechoslovak Mathematical Journal

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Let Q denote the field of rational numbers. Let K be a cyclic quartic extension of Q . It is known that there are unique integers A , B , C , D such that K = Q A ( D + B D ) , where A is squarefree and odd , D = B 2 + C 2 is squarefree , B > 0 , C > 0 , G C D ( A , D ) = 1 . The conductor f ( K ) of K is f ( K ) = 2 l | A | D , where l = 3 , if D 2 ( mod 4 ) or D 1 ( mod 4 ) , B 1 ( mod 2 ) , 2 , if D 1 ( mod 4 ) , B 0 ( mod 2 ) , A + B 3 ( mod 4 ) , 0 , if D 1 ( mod 4 ) , B 0 ( mod 2 ) , A + B 1 ( mod 4 ) . A simple proof of this formula for f ( K ) is given, which uses the basic properties of quartic Gauss sums.