Galois module structure of ideals in wildly ramified cyclic extensions of degree $p^2$

Elder Gove Griffith

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Displaying similar documents to “Galois module structure of ideals in wildly ramified cyclic extensions of degree $p^{2}$ ”

Galois module structure of ideals in wildly ramified cyclic extensions of degree $p^{2}$

Gove Griffith Elder (1995)

Annales de l'institut Fourier

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For $L / K$ , any totally ramified cyclic extension of degree $p^{2}$ of local fields which are finite extensions of the field of $p$ -adic numbers, we describe the $ℤ_{p} [Gal (L / K)]$ -module structure of each fractional ideal of $L$ explicitly in terms of the $4 p + 1$ indecomposable $ℤ_{p} [Gal (L / K)]$ -modules classified by Heller and Reiner. The exponents are determined only by the invariants of ramification.

On Galois skew polynomial rings with inner Galois groups

Szeto, George, Deng, Xinde (1990)

Portugaliae mathematica

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On Galois structure of the integers in cyclic extensions of local number fields

G. Griffith Elder (2002)

Journal de théorie des nombres de Bordeaux

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Let $p$ be a rational prime, $K$ be a finite extension of the field of $p$ -adic numbers, and let $L / K$ be a totally ramified cyclic extension of degree $p^{n}$ . Restrict the first ramification number of $L / K$ to about half of its possible values, $b_{1} > 1 / 2 \cdot p e_{0} / (p - 1)$ where $e_{0}$ denotes the absolute ramification index of $K$ . Under this loose condition, we explicitly determine the $ℤ_{p} [G]$ -module structure of the ring of integers of $L$ , where $ℤ_{p}$ denotes the $p$ -adic integers and $G$ denotes the Galois group Gal $(L / K)$ . In the process of determining...

Galois module structure of generalized jacobians.

G. D. Villa-Salvador, M. Rzedowski-Calderón (1997)

Revista Matemática de la Universidad Complutense de Madrid

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For a prime number l and for a finite Galois l-extension of function fields L / K over an algebraically closed field of characteristic p <> l, it is obtained the Galois module structure of the generalized Jacobian associated to L, l and the ramified prime divisors. In the cyclic case an implicit integral representation of the Jacobian is obtained and this representation is compared with the explicit representation.

Minkowski units in certain metacyclic fields

Robert Marszałek (1988)

Acta Arithmetica

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On the representation of functions by orthogonal series in weighted $L^{p}$ spaces

M. Grigorian (1999)

Studia Mathematica

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It is proved that if $φ_{n}$ is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form $\sum_{k = 1}^{\infty} c_{k} φ_{k} (x)$ , where $c_{k} \in l_{q}$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f \in L_{μ}^{p} [0, 1] = f : ʃ_{0}^{1} {| f (x) |}^{p} μ (x) d x < \infty$ there are numbers $ɛ_{k}$ , k=1,2,…, $ɛ_{k}$ = 1 or 0, such that $l i m_{n \to \infty} ʃ_{0}^{1} {| \sum_{k = 1}^{n} ɛ_{k} c_{k} φ_{k} (x) - f (x) |}^{p} μ (x) d x = 0 .$ 2. For every p ∈ [1,2) and $f \in L_{μ}^{p} [0, 1]$ there are a function $g \in L^{1} [0, 1]$ with g(x) = f(x) on E and numbers $δ_{k}$ , k=1,2,…, $δ_{k} = 1$ or 0,...

A note on circular units in $ℤ_{p}$ -extensions

Radan Kučera (2003)

Journal de théorie des nombres de Bordeaux

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In this note we consider projective limits of Sinnott and Washington groups of circular units in the cyclotomic $ℤ_{p}$ -extension of an abelian field. A concrete example is given to show that these two limits do not coincide in general.

On some singular integral operatorsclose to the Hilbert transform

T. Godoy, L. Saal, M. Urciuolo (1997)

Colloquium Mathematicae

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Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^{p} (ℝ)$ -boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_{m} f (x) = p . v . \int k (x - y) m (x + y) f (y) d y$ where $k (x) = \sum_{j \in ℤ} 2^{j} φ_{j} (2^{j} x)$ for a family of functions ${φ_{j}}_{j \in ℤ}$ satisfying conditions (1.1)-(1.3) given below.

Relative Galois module structure of integers of abelian fields

Nigel P. Byott, Günter Lettl (1996)

Journal de théorie des nombres de Bordeaux

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Let $L / K$ be an extension of algebraic number fields, where $L$ is abelian over $ℚ$ . In this paper we give an explicit description of the associated order $𝒜_{L / K}$ of this extension when $K$ is a cyclotomic field, and prove that $o_{L}$ , the ring of integers of $L$ , is then isomorphic to $𝒜_{L / K}$ . This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that $𝒜_{L / K}$ is the maximal order if $L / K$ is a cyclic and totally wildly ramified extension which is linearly disjoint to $ℚ^{(m^{'})} / K$ , where $m^{'}$ is the conductor...

Galois structure of ideals in wildly ramified abelian $p$ -extensions of a $p$ -adic field, and some applications

Nigel P. Byott (1997)

Journal de théorie des nombres de Bordeaux

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Let $K$ be a finite extension of $ℚ_{p}$ with ramification index $e$ , and let $L / K$ be a finite abelian $p$ -extension with Galois group $Γ$ and ramification index $p^{n}$ . We give a criterion in terms of the ramification numbers $t_{i}$ for a fractional ideal $𝔓^{h}$ of the valuation ring $S$ of $L$ not to be free over its associated order $𝔄 (K Γ; 𝔓^{h})$ . In particular, if $t_{n} - [t_{n} / p] < p^{n - 1} e$ then the inverse different can be free over its associated order only when $t_{i} \equiv - 1$ (mod $p^{n}$ ) for all $i$ . We give three consequences of this. Firstly, if $𝔄 (K Γ; S)$ is a Hopf order and...

Tauberian theorems for Cesàro summable double sequences

Ferenc Móricz (1994)

Studia Mathematica

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$(s_{j k} : j, k = 0, 1, . . .)$ be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which $(s_{j k})$ converges in Pringsheim’s sense. These conditions are satisfied if $(s_{j k})$ is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If $(s_{j k})$ is summable (C,1,1) to a finite limit and there exist constants $n_{1} > 0$ and H such that $j k (s_{j k} - s_{j - 1, k} - s_{j - 1, k} + s_{j - 1, k - 1}) \geq - H$ , $j (s_{j k} - s_{j - 1, k}) \geq - H$ and $k (s_{j k} - s_{j, k - 1}) \geq - H$ whenever $j, k > n_{1}$ , then...

Displaying similar documents to “Galois module structure of ideals in wildly ramified cyclic extensions of degree p 2 ”

Displaying similar documents to “Galois module structure of ideals in wildly ramified cyclic extensions of degree $p^{2}$ ”