Relevant bibliographies by topics / Galois ring

Academic literature on the topic 'Galois ring'

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Published: 4 June 2021
Last updated: 11 February 2022

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Journal articles on the topic "Galois ring"

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Kozlitin, Oleg A. "Estimate of the maximal cycle length in the graph of polynomial transformation of Galois–Eisenstein ring." Discrete Mathematics and Applications 28, no. 6 (December 19, 2018): 345–58. http://dx.doi.org/10.1515/dma-2018-0031.

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Abstract The paper is concerned with polynomial transformations of a finite commutative local principal ideal of a ring (a finite commutative uniserial ring, a Galois–Eisenstein ring). It is shown that in the class of Galois–Eisenstein rings with equal cardinalities and nilpotency indexes over Galois rings there exist polynomial generators for which the period of the output sequence exceeds those of the output sequences of polynomial generators over other rings.
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Szeto, George. "Separable subalgebras of a class of Azumaya algebras." International Journal of Mathematics and Mathematical Sciences 21, no. 2 (1998): 235–38. http://dx.doi.org/10.1155/s0161171298000337.

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LetSbe a ring with 1,Cthe center ofS,Ga finite automorphism group ofSof orderninvertible inS, andSGthe subnng of elements ofSfixed under each element inG. It is shown that the skew group ringS*Gis aG′-Galois extension of(S*G)G′that is a projective separableCG-algebra whereG′is the inner automorphism group ofS*Ginduced byGif and only ifSis aG-Galois extension ofSGthat is a projective separableCG-algebra. Moreover, properties of the separable subalgebras of aG-GaloisH-separable extensionSofSGare given whenSGis a projective separableCG-algebra.
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Szeto, George, and Lianyong Xue. "On central commutator Galois extensions of rings." International Journal of Mathematics and Mathematical Sciences 24, no. 5 (2000): 289–94. http://dx.doi.org/10.1155/s0161171200004099.

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LetBbe a ring with1,Ga finite automorphism group ofBof ordernfor some integern,BGthe set of elements inBfixed under each element inG, andΔ=VB(BG)the commutator subring ofBGinB. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and GaloisH-separable extensions. Several characterizations of a central commutator Galois extension are given. Moreover, it is shown that whenGis inner,Bis a central commutator Galois extension ofBGif and only ifBis anH-separable projective group ringBGGf. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer.
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Zhang, Aixian, and Keqin Feng. "Normal Bases on Galois Ring Extensions." Symmetry 10, no. 12 (December 3, 2018): 702. http://dx.doi.org/10.3390/sym10120702.

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Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.
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Szeto, George, and Linjun Ma. "On Galois projective group rings." International Journal of Mathematics and Mathematical Sciences 14, no. 1 (1991): 149–53. http://dx.doi.org/10.1155/s0161171291000145.

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LetAbe a ring with1,Cthe center ofAandG′an inner automorphism group ofAinduced by {Uαin​A/αin a finite groupGwhose order is invertible}. LetAG′be the fixed subring ofAunder the action ofG′.IfAis a Galcis extension ofAG′with Galois groupG′andCis the center of the subring∑αAG′UαthenA=∑αAG′Uαand the center ofAG′is alsoC. Moreover, if∑αAG′Uαis Azumaya overC, thenAis a projective group ring.
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Szeto, George. "On Azumaya Galois extensions and skew group rings." International Journal of Mathematics and Mathematical Sciences 22, no. 1 (1999): 91–95. http://dx.doi.org/10.1155/s0161171299220911.

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Two characterizations of an Azumaya Galois extension of a ring are given in terms of the Azumaya skew group ring of the Galois group over the extension and a Galois extension of a ring with a special Galois system is determined by the trace of the Galois group.
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PAQUES, ANTONIO, VIRGÍNIA RODRIGUES, and ALVERI SANT'ANA. "GALOIS CORRESPONDENCES FOR PARTIAL GALOIS AZUMAYA EXTENSIONS." Journal of Algebra and Its Applications 10, no. 05 (October 2011): 835–47. http://dx.doi.org/10.1142/s0219498811004999.

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Let α be a partial action, having globalization, of a finite group G on a unital ring R. Let Rα denote the subring of the α-invariant elements of R and CR(Rα) the centralizer of Rα in R. In this paper we will show that there are one-to-one correspondences among sets of suitable separable subalgebras of R, Rα and CR(Rα). In particular, we extend to the setting of partial group actions similar results due to DeMeyer [Some notes on the general Galois theory of rings, Osaka J. Math.2 (1965) 117–127], and Alfaro and Szeto [On Galois extensions of an Azumaya algebra, Commun. Algebra25 (1997) 1873–1882].
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Mináč, Ján, and Tara L. Smith. "Decomposition of Witt Rings and Galois Groups." Canadian Journal of Mathematics 47, no. 6 (December 1, 1995): 1274–89. http://dx.doi.org/10.4153/cjm-1995-065-0.

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AbstractTo each field F of characteristic not 2, one can associate a certain Galois group 𝒢F, the so-called W-group of F, which carries essentially the same information as the Witt ring W(F) of F. In this paper we show that direct products of Witt rings correspond to free products of these Galois groups (in the appropriate category), group ring construction of Witt rings corresponds to semidirect products of W-groups, and the basic part of W(F) is related to the center of 𝒢F. In an appendix we provide a complete list of Witt rings and corresponding w-groups for fields F with |Ḟ/Ḟ2| ≤ 16.
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Jiang, Xiaolong, and George Szeto. "The Galois endomorphism ring of a Galois Azumaya extension." International Journal of Algebra 7 (2013): 909–14. http://dx.doi.org/10.12988/ija.2013.311111.

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Cao, Yonglin. "Generalized affine transformation monoids on Galois rings." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–6. http://dx.doi.org/10.1155/ijmms/2006/90738.

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LetAbe a ring with identity. The generalized affine transformation monoidGaff(A)is defined as the set of all transformations onAof the formx↦xu+a(for allx∈A), whereu,a∈A. We study the algebraic structure of the monoidGaff(A)on a finite Galois ringA. The following results are obtained: an explicit description of Green's relations onGaff(A); and an explicit description of the Schützenberger group of every-class, which is shown to be isomorphic to the affine transformation group for a smaller Galois ring.
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Dissertations / Theses on the topic "Galois ring"

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Hedenlund, Alice. "Galois Theory of Commutative Ring Spectra." Thesis, KTH, Matematik (Avd.), 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-183512.

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This thesis discusses Galois theory of ring spectra in the sense of John Rognes. The aim is to give a clear introduction that provides a solid foundation for further studies into the subject. We introduce ring spectra using the symmetric spectra of Hovey, Shipley and Smith, and discuss the symmetric monoidal model structure on this category. We define and give results for Galois extensions of these objects. We also give examples involving Eilenberg-Mac Lane spectra of commutative rings, topological K-theory spectra and cochain algebras of these. Galois extensions of ring spectra are compared to Ga-lois extensions of commutative rings especially relating to faithfulness, a property that is implicit in the latter, but not in the former. This is proven by looking at extensions of cochain algebras using Eilenberg-Mac Lane spectra. We end by contrasting this to cochain algebra extensions using K-theory spectra, and show that such extensions are not Galois, using methods of Baker and Richter.
Denna uppsats behandlar Galoisutvidgningar av ringspektra som först introducerade av Rognes. Målet är att ge en klar introduktion för en sta-bil grund för vidare studier inom ämnet. Vi introducerar ringspektra genom att använda oss av symmetris-ka spektra utvecklade av Hovey, Shipley och Smith, och diskuterar den symmetriskt monoidiala modelstrukturen på denna kategori. Vi definierar och ger resultat för Galoisutvidgningar av dessa objekt. Vi ger också en mängd exempel, som till exempel utvidgningar av Eilenberg-Mac Lane spektra av kommutativa ringar, topologiska K-teorispektra och koked-jealgebror. Galoisutvidgningar av ringspektra jämförs med Galoisutvidgningar av kommutativa ringar, speciellt med avseende pa˚ trogenhet, en egenskap som ¨ar en inneboende egenskap hos den senare men inte i den förra. Detta visas genom att betrakta utvidgningar av kokedjealgebror av Eilenberg-Mac Lane spektra. Vi avslutar med att jämföra detta med kokedjealgebrautvidgningar av K-teorispektra och visar att sådana inte är Galois genom att använda metoder utvecklade av Baker och Richter
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Abrahamsson, Björn. "Architectures for Multiplication in Galois Rings." Thesis, Linköping University, Department of Electrical Engineering, 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-2396.

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This thesis investigates architectures for multiplying elements in Galois rings of the size 4^m, where m is an integer.

The main question is whether known architectures for multiplying in Galois fields can be used for Galois rings also, with small modifications, and the answer to that question is that they can.

Different representations for elements in Galois rings are also explored, and the performance of multipliers for the different representations is investigated.

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Armand, Marc Andre. "Contributions to the decoding of linear codes over a Galois ring." Thesis, University of Bristol, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.297851.

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Moon, Yong Suk. "Galois Deformation Ring and Barsotti-Tate Representations in the Relative Case." Thesis, Harvard University, 2016. http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493581.

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In this thesis, we study finite locally free group schemes, Galois deformation rings, and Barsotti-Tate representations in the relative case. We show three independent but related results, assuming p > 2. First, we give a simpler alternative proof of Breuil’s result on classifying finite flat group schemes over the ring of integers of a p-adic field by certain Breuil modules [5]. Second, we prove that the locus of potentially semi-stable representations of the absolute Galois group of a p-adic field K with a specified Hodge-Tate type and Galois type cuts out a closed subspace of the generic fiber of a given Galois deformation ring, without assuming that K/Qp is finite. This is an extension of the corresponding result of Kisin when K/Qp is finite [19]. Third, we study the locus of Barsotti-Tate representations in the relative case, via analyzing certain extendability of p-divisible groups. We prove that when the ramification index is less than p-1, the locus of relative Barsotti-Tate representations cuts out a closed subspace of the generic fiber of a Galois deformation ring, if the base scheme is 2-dimensional satisfying some conditions. When the ramification index is greater than p-1, we show that such a result does not hold in general.
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Szabo, Steve. "Convolutional Codes with Additional Structure and Block Codes over Galois Rings." Ohio University / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1257792383.

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Pinnawala, Nimalsiri, and nimalsiri pinnawala@rmit edu au. "Properties of Trace Maps and their Applications to Coding Theory." RMIT University. Mathematical and Geospatial Sciences, 2008. http://adt.lib.rmit.edu.au/adt/public/adt-VIT20080515.121603.

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In this thesis we study the application of trace maps over Galois fields and Galois rings in the construction of non-binary linear and non-linear codes and mutually unbiased bases. Properties of the trace map over the Galois fields and Galois rings has been used very successfully in the construction of cocyclic Hadamard, complex Hadamard and Butson Hadamard matrices and consequently to construct linear codes over integers modulo prime and prime powers. These results provide motivation to extend this work to construct codes over integers modulo . The prime factorization of integers paved the way to focus our attention on the direct product of Galois rings and Galois fields of the same degree. We define a new map over the direct product of Galois rings and Galois fields by using the usual trace maps. We study the fundamental properties of the this map and notice that these are very similar to that of the trace map over Galois rings and Galois fields. As such this map called the trace-like map and is used to construct cocyclic Butson Hadamard matrices and consequently to construct linear codes over integers modulo . We notice that the codes construct in this way over the integers modulo 6 is simplex code of type . A further generalization of the trace-like map called the weighted-trace map is defined over the direct product of Galois rings and Galois fields of different degrees. We use the weighted-trace map to construct some non-linear codes and mutually unbiased bases of odd integer dimensions. Further more we study the distribution of over the Galois fields of degree 2 and use it to construct 2-dimensional, two-weight, self-orthogonal codes and constant weight codes over integers modulo prime.
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Herschend, Martin. "On the Clebsch-Gordan problem for quiver representations." Doctoral thesis, Uppsala University, Department of Mathematics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-8663.

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On the category of representations of a given quiver we define a tensor product point-wise and arrow-wise. The corresponding Clebsch-Gordan problem of how the tensor product of indecomposable representations decomposes into a direct sum of indecomposable representations is the topic of this thesis.

The choice of tensor product is motivated by an investigation of possible ways to modify the classical tensor product from group representation theory to the case of quiver representations. It turns out that all of them yield tensor products which essentially are the same as the point-wise tensor product.

We solve the Clebsch-Gordan problem for all Dynkin quivers of type A, D and E6, and provide explicit descriptions of their respective representation rings. Furthermore, we investigate how the tensor product interacts with Galois coverings. The results obtained are used to solve the Clebsch-Gordan problem for all extended Dynkin quivers of type Ãn and the double loop quiver with relations βα=αβ=αn=βn=0.

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Lodh, Rémi Shankar. "Galois cohomology of Fontaine rings." [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=984696563.

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Turner, Ryan. "The Galois theory of matrix C-rings." Thesis, Swansea University, 2006. https://cronfa.swan.ac.uk/Record/cronfa42607.

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Blackford, J. Thomas. "Permutation groups of extended cyclic codes over Galois Rings /." The Ohio State University, 1999. http://rave.ohiolink.edu/etdc/view?acc_num=osu1488186329502909.

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Books on the topic "Galois ring"

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Greither, Cornelius. Cyclic Galois extensions of commutative rings. Berlin: Springer-Verlag, 1992.

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Rognes, John. Galois extensions of structured ring spectra: Stably dualizable groups. Providence, R.I: American Mathematical Society, 2008.

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Zbigniew, Hajto, ed. Algebraic groups and differential Galois theory. Providence, R.I: American Mathematical Society, 2011.

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Snaith, Victor P. Groups, rings and Galois theory. 2nd ed. Singapore: World Scientific, 2004.

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Groups, rings and Galois theory. 2nd ed. River Edge, N.J: World Scientific, 2003.

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1912-, Milgram Arthur N., ed. Galois theory. Mineola, N.Y: Dover Publications, 1998.

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S, Passman Donald. Group rings, crossed products, and Galois theory. Providence, R.I: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 1986.

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Greither, Cornelius. Cyclic Galois Extensions of Commutative Rings. Berlin, Heidelberg: Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/bfb0089165.

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Lectures on finite fields and galois rings. Singapore: World Scientific, 2003.

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Lectures on finite fields and galois rings. Singapore: World Scientific, 2004.

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Book chapters on the topic "Galois ring"

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Childs, Lindsay N. "On Hopf Galois extensions of fields and number rings." In Perspectives in Ring Theory, 117–28. Dordrecht: Springer Netherlands, 1988. http://dx.doi.org/10.1007/978-94-009-2985-2_11.

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Byrne, Eimear. "Lifting Decoding Schemes over a Galois Ring." In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 323–32. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/3-540-45624-4_34.

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Nechaev, Alexander A., and Alexey S. Kuzmin. "Trace-function on a Galois ring in coding theory." In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 277–90. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/3-540-63163-1_22.

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Dinh, Hai Q. "On Some Classes of Repeated-root Constacyclic Codes of Length a Power of 2 over Galois Rings." In Advances in Ring Theory, 131–47. Basel: Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-0346-0286-0_10.

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Tsypyschev, V. N. "Lower Bounds on Linear Complexity of Digital Sequences Products of LRS and Matrix LRS over Galois Ring." In Advances in Intelligent Systems and Computing, 50–61. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-67618-0_6.

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Bini, Gilberto, and Flaminio Flamini. "Galois and Quasi-Galois Rings: Structure and Properties." In Finite Commutative Rings and Their Applications, 81–119. Boston, MA: Springer US, 2002. http://dx.doi.org/10.1007/978-1-4615-0957-8_6.

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Bini, Gilberto, and Flaminio Flamini. "Galois Theory for Local Rings." In Finite Commutative Rings and Their Applications, 71–80. Boston, MA: Springer US, 2002. http://dx.doi.org/10.1007/978-1-4615-0957-8_5.

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Kaur, Jasbir, Sucheta Dutt, and Ranjeet Sehmi. "Cyclic Codes over Galois Rings." In Algorithms and Discrete Applied Mathematics, 233–39. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-29221-2_20.

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Greither, Cornelius. "Galois theory of commutative rings." In Cyclic Galois Extensions of Commutative Rings, 1–31. Berlin, Heidelberg: Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/bfb0089166.

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Fenrick, Maureen H. "Preliminaries — Groups and Rings." In Introduction to the Galois Correspondence, 1–72. Boston, MA: Birkhäuser Boston, 1992. http://dx.doi.org/10.1007/978-1-4684-0026-7_1.

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Conference papers on the topic "Galois ring"

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Akleyleky, Sedat, and Ferruh Ozbudak. "Multiplication in a Galois ring." In 2015 Seventh International Workshop on Signal Design and its Applications in Communications (IWSDA). IEEE, 2015. http://dx.doi.org/10.1109/iwsda.2015.7458407.

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Boztas, Serdar, and Parampalli Udaya. "Partial Correlations of Galois Ring Sequences." In 2007 3rd International Workshop on Signal Design and its Applications in Communications. IEEE, 2007. http://dx.doi.org/10.1109/iwsda.2007.4408347.

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May, J. P. "What precisely are E∞ring spaces and E∞ring spectra?" In New topological contexts for Galois theory and algebraic geometry. Mathematical Sciences Publishers, 2009. http://dx.doi.org/10.2140/gtm.2009.16.215.

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May, J. P. "What are E∞ring spaces good for?" In New topological contexts for Galois theory and algebraic geometry. Mathematical Sciences Publishers, 2009. http://dx.doi.org/10.2140/gtm.2009.16.331.

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Schlichtkrull, Christian. "The cyclotomic trace for symmetric ring spectra." In New topological contexts for Galois theory and algebraic geometry. Mathematical Sciences Publishers, 2009. http://dx.doi.org/10.2140/gtm.2009.16.545.

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May, J. P. "The construction of E∞ring spaces from bipermutative categories." In New topological contexts for Galois theory and algebraic geometry. Mathematical Sciences Publishers, 2009. http://dx.doi.org/10.2140/gtm.2009.16.283.

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May, J. P. "The construction of E∞ring spaces from bipermutative categories." In New topological contexts for Galois theory and algebraic geometry. Mathematical Sciences Publishers, 2009. http://dx.doi.org/10.2140/gtm.2009.16.285.

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Demir, Kaya, and Salih Ergun. "Analysis of Random Number Generators Based on Fibonacci-Galois Ring Oscillators." In 2019 17th IEEE International New Circuits and Systems Conference (NEWCAS). IEEE, 2019. http://dx.doi.org/10.1109/newcas44328.2019.8961262.

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Garcia-Bosque, M., G. Diez-Senorans, C. Sanchez-Azqueta, and S. Celma. "FPGA Implementation of a New PUF Based on Galois Ring Oscillators." In 2021 IEEE 12th Latin America Symposium on Circuits and System (LASCAS). IEEE, 2021. http://dx.doi.org/10.1109/lascas51355.2021.9459135.

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Demir, Kaya, and Salih Ergun. "A Comparative Study on Fibonacci-Galois Ring Oscillators for Random Number Generation." In 2020 IEEE 63rd International Midwest Symposium on Circuits and Systems (MWSCAS). IEEE, 2020. http://dx.doi.org/10.1109/mwscas48704.2020.9184524.

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Reports on the topic "Galois ring"

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Ashikhmin, A. On generalized hamming weighs for Galois ring linear codes. Office of Scientific and Technical Information (OSTI), August 1997. http://dx.doi.org/10.2172/515638.

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