On the $n$-torsion subgroup of the Brauer group of a number field

Hershy Kisilevsky; Jack Sonn

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Boundedness results of solutions to the equation $x^{\prime\prime\prime}+ax^{\prime\prime}+g(x)x^{\prime}+h(x)=p(t)$ without the hypothesis $h(x)\,\text{sgn}x\geq 0$ $for|x|>R$ .

Ján Andres (1986)

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

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Per l'equazione differenziale ordinaria non lineare del 3° ordine indicata nel titolo, studiata da numerosi autori sotto l'ipotesi $h(x)\,\text{sgn}x\geq 0$ $for|x|>R$ , si dimostra l'esistenza di almeno una soluzione limitata sopprimendo l'ipotesi suddetta.

On the ring of $p$ -integers of a cyclic $p$ -extension over a number field

Humio Ichimura (2005)

Journal de Théorie des Nombres de Bordeaux

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Let $p$ be a prime number. A finite Galois extension $N / F$ of a number field $F$ with group $G$ has a normal $p$ -integral basis ( $p$ -NIB for short) when $𝒪_{N}^{'}$ is free of rank one over the group ring $𝒪_{F}^{'} [G]$ . Here, $𝒪_{F}^{'} = 𝒪_{F} [1 / p]$ is the ring of $p$ -integers of $F$ . Let $m = p^{e}$ be a power of $p$ and $N / F$ a cyclic extension of degree $m$ . When $ζ_{m} \in F^{\times}$ , we give a necessary and sufficient condition for $N / F$ to have a $p$ -NIB (Theorem 3). When $ζ_{m} \notin F^{\times}$ and $p ∤ [F (ζ_{m}) : F]$ , we show that $N / F$ has a $p$ -NIB if and only if $N (ζ_{m}) / F (ζ_{m})$ has a $p$ -NIB (Theorem 1). When $p$ divides $[F (ζ_{m}) : F]$ , we show that this...

Boundedness results of solutions to the equation $x^{\prime\prime\prime}+ax^{\prime\prime}+g(x)x^{\prime}+h(x)=p(t)$ without the hypothesis $h(x)\,\operatorname{sgn}x\geq 0$ for $|x|>R$ .

Ján Andres (1986)

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

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Quaternion extensions with restricted ramification

Peter Schmid (2014)

Acta Arithmetica

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In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime $p \equiv 1 (m o d 2^{n - 1})$ , there exist unique real and complex...

On sets which contain a qth power residue for almost all prime modules

Mariusz Ska/lba (2005)

Colloquium Mathematicae

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A classical theorem of M. Fried [2] asserts that if non-zero integers $β ₁, . . ., β_{l}$ have the property that for each prime number p there exists a quadratic residue $β_{j}$ mod p then a certain product of an odd number of them is a square. We provide generalizations for power residues of degree n in two cases: 1) n is a prime, 2) n is a power of an odd prime. The proofs involve some combinatorial properties of finite Abelian groups and arithmetic results of [3].

On blocks of arithmetic progressions with equal products

N. Saradha (1997)

Journal de théorie des nombres de Bordeaux

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Let $f (X) \in ℚ [X]$ be a monic polynomial which is a power of a polynomial $g (X) \in ℚ [X]$ of degree $μ \geq 2$ and having simple real roots. For given positive integers $d_{1}, d_{2}, ℓ, m$ with $ℓ < m$ and gcd $(ℓ, m) = 1$ with $μ \leq m + 1$ whenever $m < 2$ , we show that the equation $f (x) f (x + d_{1}) \dots f (x + (ℓ k - 1) d_{1}) = f (y) f (y + d_{2}) \dots f (y + (m k - 1) d_{2})$ with $f (x + j d_{1}) \neq 0$ for $0 \leq j < ℓ k$ has only finitely many solutions in integers $x, y$ and $k \geq 1$ except in the case $m = μ = 2, ℓ = k = d_{2} = 1, f (X) = g (X), x = f (y) + y .$

Making sense of capitulation: reciprocal primes

David Folk (2016)

Acta Arithmetica

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Let ℓ be a rational prime, K be a number field that contains a primitive ℓth root of unity, L an abelian extension of K whose degree over K, [L:K], is divisible by ℓ, a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and $a_{}$ any generator of the principal ideal $^{ℓ}$ . We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates $G a l (K (\sqrt[ℓ]{a_{}}) / K)$ for every choice of $a_{}$ . We then show that becomes principal in L if and only if every reciprocal prime is not...

Infinitude of Wilson primes for $_{q} [t]$

Jim Sauerberg, Linghsueh Shu, Dinesh S. Thakur, George Todd (2013)

Acta Arithmetica

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Galois co-descent for étale wild kernels and capitulation

Manfred Kolster, Abbas Movahhedi (2000)

Annales de l'institut Fourier

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Let $F$ be a number field with ring of integers $o_{F}$ . For a fixed prime number $p$ and $i \geq 2$ the étale wild kernels $W K_{2 i - 2}^{\overset{´}{e} t} (F)$ are defined as kernels of certain localization maps on the $i$ -fold twist of the $p$ -adic étale cohomology groups of $spec o_{F} [\frac{1}{p}]$ . These groups are finite and coincide for $i = 2$ with the $p$ -part of the classical wild kernel $W K_{2} (F)$ . They play a role similar to the $p$ -part of the $p$ -class group of $F$ . For class groups, Galois co-descent in a cyclic extension $L / F$ is described by the ambiguous class formula given...

Pure fields of degree 9 with class number prime to 3

Colin D. Walter (1980)

Annales de l'institut Fourier

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The main theorem gives necessary conditions and sufficient conditions for $Q (\sqrt[9]{n})$ to have class number prime to 3. These conditions involve only the rational prime factorization of $n$ and congruences mod 27 of the prime factors of $n$ . They give necessary and sufficient conditions for most $n$ .

Displaying similar documents to “On the n -torsion subgroup of the Brauer group of a number field”

Displaying similar documents to “On the $n$ -torsion subgroup of the Brauer group of a number field”