Galois coverings and splitting properties of the ideal generated by halflines

Piotr Dowbor

Displaying similar documents to “Galois coverings and splitting properties of the ideal generated by halflines”

Two classes of almost Galois coverings for algebras

Piotr Dowbor, Adam Hajduk (2012)

Colloquium Mathematicae

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We prove that for any representation-finite algebra A (in fact, finite locally bounded k-category), the universal covering F: Ã → A is either a Galois covering or an almost Galois covering of integral type, and F admits a degeneration to the standard Galois covering F̅: Ã→ Ã/G, where $G = Π (Γ_{A})$ is the fundamental group of $Γ_{A}$ . It is shown that the class of almost Galois coverings F: R → R’ of integral type, containing the series of examples from our earlier paper [Bol. Soc. Mat. Mexicana 17 (2011)],...

Non-orbicular modules for Galois coverings

Piotr Dowbor (2001)

Colloquium Mathematicae

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Given a group G of k-linear automorphisms of a locally bounded k-category R, the problem of existence and construction of non-orbicular indecomposable R/G-modules is studied. For a suitable finite sequence B of G-atoms with a common stabilizer H, a representation embedding $Φ^{B} : I ₙ - s p r (H) \to m o d (R / G)$ , which yields large families of non-orbicular indecomposable R/G-modules, is constructed (Theorem 3.1). It is proved that if a G-atom B with infinite cyclic stabilizer admits a non-trivial left Kan extension B̃ with...

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...

On Galois $H$ -objects and invertible $H^{*}$ -modules

J. N. Alonso Alvarez, J. M. Fernández Vilaboa (2000)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

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Polynomials over $Q$ solving an embedding problem

Nuria Vila (1985)

Annales de l'institut Fourier

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The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group $A_{n}$ , can be embedded in any central extension of $A_{n}$ if and only if $n \equiv 0 (m o d 8)$ , or $n \equiv 2 (m o d 8)$ and $n$ is a sum of two squares. Consequently, for theses values of $n$ , every central extension of $A_{n}$ occurs as a Galois group over $Q$ .

Differential Galois Theory for an Exponential Extension of $ℂ ((z))$

Magali Bouffet (2003)

Bulletin de la Société Mathématique de France

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In this paper we study the formal differential Galois group of linear differential equations with coefficients in an extension of $ℂ ((z))$ by an exponential of integral. We use results of factorization of differential operators with coefficients in such a field to give explicit generators of the Galois group. We show that we have very similar results to the case of $ℂ ((z))$ .

On a notion of “Galois closure” for extensions of rings

Manjul Bhargava, Matthew Satriano (2014)

Journal of the European Mathematical Society

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We introduce a notion of “Galois closure” for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an $S_{n}$ degree $n$ extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions.

Some remarks on Hilbert-Speiser and Leopoldt fields of given type

James E. Carter (2007)

Colloquium Mathematicae

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Let p be a rational prime, G a group of order p, and K a number field containing a primitive pth root of unity. We show that every tamely ramified Galois extension of K with Galois group isomorphic to G has a normal integral basis if and only if for every Galois extension L/K with Galois group isomorphic to G, the ring of integers $O_{L}$ in L is free as a module over the associated order $_{L / K}$ . We also give examples, some of which show that this result can still hold without the assumption that...

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...

The Galois group of $X^{p} + a X^{s} + a$

B. Bensebaa, A. Movahhedi, A. Salinier (2008)

Acta Arithmetica

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Galois module structure of the rings of integers in wildly ramified extensions

Stephen M. J. Wilson (1989)

Annales de l'institut Fourier

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The main results of this paper may be loosely stated as follows. Theorem.— Let $N$ and $N^{'}$ be sums of Galois algebras with group $Γ$ over algebraic number fields. Suppose that $N$ and $N^{'}$ have the same dimension $ℚ$ and that they are identical at their wildly ramified primes. Then (writing $𝒪_{N}$ for the maximal order in $N$ ) $𝒪_{N} \oplus 𝒪_{N} \oplus ℤ Γ ≅_{ℤ Γ} \oplus 𝒪_{N}^{'} \oplus 𝒪_{N^{'}} \oplus ℤ Γ .$

Differential Galois realization of double covers

Teresa Crespo, Zbigniew Hajto (2002)

Annales de l’institut Fourier

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An effective construction of homogeneous linear differential equations of order 2 with Galois group $2 A_{4}, 2 S_{4}$ or $2 A_{5}$ is presented.