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"About Galois Theory and Applications – Solved Exercises and Problems ",
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"by Mohamed Ayad"
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"Reviewed by Jean-Paul Allouche"
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"A book entirely devoted to exercises and problems with solutions about Galois\ntheory and its applications: is this something new? is this something\ninteresting? The answer to the first question is probably yes. Of course there\nare already several books on the subject that contain exercises; there is even\na book entitled ",
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"content": [
"Galois Theory Through Exercises"
]
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" by J. Brzeziński. But\nthe book under review contains nothing but a very large number of exercises\n(285) that one ",
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"type": "Emphasis",
"content": [
"irresistibly"
]
},
" wants to attack – without even looking at\nthe solutions provided by the author. That is what I did, browsing through the\neleven chapters of the book (Polynomials, Fields, Generalities; Algebraic\nextension, Algebraic closure; Separability, Inseparability; Normal extensions,\nGalois extensions, Galois groups; Finite fields; Permutation polynomials;\nTranscendental extensions, Linearly disjoint extensions, Luroth’s theorem;\nMultivariate polynomials; Integral elements, Algebraic number theory;\nDerivations), picking exercises, and trying to solve them. Here are a few\nexamples: Exercise 2.31 begins with solving ",
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", and ends by asking for a proof that ",
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" can be expressed as the composition of two polynomials of degree ",
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".\nExercise 5.26 plays with the cyclic extensions of degree ",
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" over the\nrationals; Exercise 8.1 looks innocent but is related to the Christol theorem\non algebraicity of “automatic” series; Exercise 8.3 requires proving (with a\nhint …) that if ",
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", where ",
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" is an extension of finite type of an algebraically closed field\n",
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", then one must have ",
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". Exercise 10.24 leads to the\ndetermination of all the distinct factorizations of ",
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" into a product of\nirreducibles in ",
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", a problem which yields the nice\nimage of the cover page. Solving (or trying to solve …) all the exercises\ngives a clear answer to the second question at the beginning of this survey:\nyes, this book is definitely interesting, and I warmly recommend it. It can be\nused by beginners who want to learn about Galois theory in a more playful\nmanner, by colleagues who want to teach really everything about Galois theory,\nand even by researchers who might discover useful results and ideas there."
]
},
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"content": [
"Mohamed Ayad,\n",
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"content": [
"About Galois Theory and Applications. Solved Exercises and Problems"
]
},
".\nWorld Scientific, 2018, 452 pages.\nHardcover ISBN 978-981-3238-30-5.\neBook ISBN 978-981-3238-32-9.\n\n"
]
},
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"type": "Heading",
"id": "Sx2",
"depth": 1,
"content": [
"Meilensteine der Rechentechnik (Milestones in Analog and Digital Computing) ",
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"content": [
"by Herbert Bruderer"
]
}
]
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"Reviewed by Jean-Paul Allouche"
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"After the impressive and remarkable first edition (one volume of more than\n800 pages, see the ",
{
"type": "Emphasis",
"content": [
"Newsletter of the EMS"
]
},
", December 2016, Issue 102,\np. 154), this second edition consists of two volumes totaling over 1500 pages.\nIt has the same good qualities as the previous edition but contains twice as\nmuch material, which makes the set of the two volumes of the second edition an\nextremely useful contribution to the history of computing machines. I will\nconcentrate on the second volume. The first part is devoted to general\nquestions and answers about computers, from “Who invented the first computer”\nto “What is a Turing machine?” through questions about theoretical computer\nscience, algorithms and universal machines; from “What is a von Neumann\ncomputer” (and is it a series or parallel computer?) to theoretical questions\nabout storage; from political and historical issues to technical developments.\nThis first chapter is already extraordinarily rich. It is followed by chapters\nthat provide a detailed analysis of events in three different countries, namely\nGermany, Great Britain, and Switzerland. In these chapters, we learn an\nincredible number of things that most of us probably never suspected, e.g., the\ndifference between computing machines and logical devices according to Konrad\nZuse (and questions about “computers” playing chess), the whole history of\n",
{
"type": "Emphasis",
"content": [
"Enigma"
]
},
" and of the “Turing–Welchman bomb”, the question of whether\nChurchill really ordered all “colossal” computers to be destroyed, the\nhistory of the Swiss computer ",
{
"type": "Emphasis",
"content": [
"Ermeth"
]
},
", acronym for “Elektronische\nRechenmaschine der ETH” (followed in particular by ",
{
"type": "Emphasis",
"content": [
"Lilith"
]
},
" and\n",
{
"type": "Emphasis",
"content": [
"Ceres"
]
},
": given that these last two names are related to religious figures,\none might ask whether the acronym Ermeth had something to do with the Hebrew\nword for truth, namely ",
{
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"content": [
"emeth"
]
},
", see, e.g.,\n",
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",\npage 50). A further chapter is devoted to the first computing devices from\nalmost twenty other countries: let me just cite Spain with Leonardo Torres y\nQuevedo, his analog computer, his chess-playing computer and his analytical\nmachine. The book also contains an extremely useful dictionary for all\ntechnical terms, giving the English equivalent for all German words or\nexpressions and vice versa, not to mention a bibliography over 300 pages long\nand an amazing set of images! After having read this volume and the first\nvolume, what strikes me most is the incredibly rich history of computer\nscience, and the incredibly deep ignorance of this history by essentially\neverybody who uses computers for whatever purpose. This is just one of the\nreasons for which these two beautiful and well-documented volumes should\ndefinitely be necessary reading."
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"content": [
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": I was going to add that it would be good to translate these\nvolumes into English (and French), when I learned that an English version is\ndue to appear very soon – the electronic English version was made available on\nJanuary 6\n(",
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"\nHerbert Bruderer, ",
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"Meilensteine der Rechentechnik. Erfindung des Computers, Elektronenrechner, Entwicklungen in Deutschland, England und der Schweiz"
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},
", 2. Auflage, Band 2\n(",
{
"type": "Emphasis",
"content": [
"Milestones in Analog and Digital Computing"
]
},
", 2nd edition, volume 2).\nDe Gruyter Oldenburg, 2018, 829 pages.\nISBN 978-3-11-060088-9. e-ISBN 978-3-11-060261-6.\n"
]
},
{
"type": "Paragraph",
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"content": [
"\nJean-Paul Allouche is “Directeur de recherche” emeritus, at CNRS. He is\nworking at IMJ-PRG, Sorbonne, Paris (France) on subjects relating number theory\nand theoretical computer science, including the so-called “automatic\nsequences”. He was editor of the ",
{
"type": "Emphasis",
"content": [
"EMS Newsletter"
]
},
" in 2013–2020.\n\n",
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"id": "Sx2.p5",
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"id": "Sx3",
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"Recurrent Sequences. Key Results, Applications and Problems\n",
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"by Dorin Andrica and Ovidiu Bagdasar"
]
}
]
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"Reviewed by Michael Th. Rassias"
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"This book contains an ample presentation of recurrent sequences from multiple\nperspectives, initiating the readers with classical results and gradually\nleading them to the very frontier of what is known in the subject. The\nexpository style is engaging and the succinct presentation of theoretical\nresults is accompanied by short but tricky examples which invite the reader to\ninvestigate the topic further."
]
},
{
"type": "Paragraph",
"id": "Sx3.p4",
"content": [
"The first six chapters of the book present classical and recent results on the\ntopic. Numerous results have been obtained by the authors, and highlight\nconnections between recurrences and combinatorics, number theory, integer\nsequences, and random number generation. The diagrams of orbits of second and\nthird-order recurrent sequences in the complex plane presented in the book add\nsignificantly to its artistic quality. About a third of the book is devoted to\nan inspired selection of 123 (the 10th Lucas number) Olympiad training\nproblems, accompanied by detailed solutions."
]
},
{
"type": "Paragraph",
"id": "Sx3.p5",
"content": [
"Chapter 1 offers a succinct presentation of the fundamentals of recurrence\nrelations, along with examples of recurrent sequences naturally arising in\nalgebra, combinatorics, geometry, analysis, and mathematical modelling."
]
},
{
"type": "Paragraph",
"id": "Sx3.p6",
"content": [
"Chapter 2 is devoted to first and second-order linear recursions, as well as\nhomographic recurrences. Examples include the Fibonacci sequence and its close\ncompanions: the Lucas, Pell or Pell–Lucas sequences, for which the authors\npresent a palette of interesting identities with elegant proofs. The\ndiscussion extends to special families of polynomials, which are then related\nback to the Fibonacci, Lucas, Pell and Pell–Lucas sequences, and used to\nestablish novel number theoretic results. This chapter also presents\nhomographic sequences with constant and variable coefficients."
]
},
{
"type": "Paragraph",
"content": [
"Chapter 3 presents arithmetic properties of the Fibonacci, Lucas, Pell and\nPell–Lucas sequences, with links to pseudoprimality. The authors prove new\ntheoretical results, present recent entries to the Online Encyclopedia of\nInteger Sequences, and formulate a few interesting conjectures. Some of these\nresults have already been extended to generalized Pell and Pell-Lucas sequences\nin the recent paper [Andrica, D. and Bagdasar, O.: On some new arithmetic\nproperties of the generalized Lucas sequences. ",
{
"type": "Emphasis",
"content": [
"Mediterr. J. Math."
]
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", to appear\n(2021)]. The complex factorization of the polynomials"
]
},
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"is used to derive some elegant trigonometric formulae for these classical\nsequences."
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{
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"content": [
"Chapter 4 is devoted to ordinary and exponential generating functions, which\nare used to evaluate the general term formulae for many classical polynomials\nand sequences in Sections 4.1 and 4.2. The interesting version of Cauchy’s\nintegral formula given in Section 4.3 is used to derive integral formulae for\nthe Fibonacci, Lucas, Pell and Pell–Lucas sequences. In some recent papers the\nauthors have used this approach to establish novel integral formulae for the\ncoefficients of cyclotomic, Gaussian, multinomial, or polygonal polynomials."
]
},
{
"type": "Paragraph",
"id": "Sx3.p9",
"content": [
"Chapter 5 explores second order linear recurrences depending on a family of\nfour complex coefficients (often called Horadam sequences). The results include\nnecessary and sufficient conditions for periodicity (Section 5.2), the\ngeometric structure (Section 5.3), and the enumeration of Horadam orbits with a\nfixed length (Section 5.4). An atlas presenting numerous beautiful diagrams of\nperiodic and non-periodic Horadam patterns is given in Section 5.6, while\nSection 5.7 presents a Horadam-based pseudo-random number generator. Some\nexamples of periodic non-homogeneous Horadam sequences are given in\nSection 5.8. The chapter is based on many recent articles."
]
},
{
"type": "Paragraph",
"id": "Sx3.p10",
"content": [
"Chapter 6 further develops the ideas presented in Chapter 2, featuring a\ncollection of useful methods related to generating functions, matrices and\ninterpolating geometric inequalities. Some results for systems of linear\nrecurrence sequences are also given, with applications to Diophantine\nequations. Extending the results from Chapter 5, the authors present complex\nlinear recurrent sequences of higher order, periodicity conditions, geometric\nstructure, and enumeration of periodic orbits with a fixed length. An atlas of\nexciting geometric patterns produced by third-order linear recursions in the\ncomplex plane is also showcased. The chapter concludes with a presentation of\nconnections between the theory of linear recurrences and finite differences."
]
},
{
"type": "Paragraph",
"id": "Sx3.p11",
"content": [
"Chapter 7 contains 123 Olympiad training problems involving recurrent\nsequences, which are solved in detail in Chapter 8, sometimes with multiple\nsolutions. The problems concern linear recurrence sequences of first, second\nand higher orders, some classical sequences, homographic sequences, systems of\nsequences, complex recurrence sequences, and recursions in combinatorics. Many\nof the problems were actually proposed by the authors, while the others were\nselected from international competitions or classical journals."
]
},
{
"type": "Paragraph",
"id": "Sx3.p12",
"content": [
"The book ends with an appendix and a rich bibliography including\n177 references, many of which represent contributions by the authors. An index\nis also provided."
]
},
{
"type": "Paragraph",
"id": "Sx3.p13",
"content": [
"This book teaches numerous fundamental facts and techniques which are central\nin mathematics. It is both a research monograph and a delightful problem book,\nwhich I feel will spark the interest of a wide audience, from mathematics\nOlympiad competitors and their coaches to undergraduate or postgraduate\nstudents, or professional mathematicians with an interest in recurrences and\ntheir multiple applications."
]
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"\nDorin Andrica and Ovidiu Bagdasar,\n",
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",\nSpringer International Publishing, 2020, 402 + xiv pages.\nHardcover ISBN 978-3-030-51501-0.\neBook ISBN 978-3-030-51502-7.\n"
]
},
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"\nMichael Th. Rassias is a member of the\nEditorial Board of the Newsletter/Magazine of the EMS. He is a Research\nFellow at the I-Math of the University of Zürich and a visiting\nresearcher at PIDS of the IAS, Princeton. He holds a Diploma from the\nNTUA, Greece, a Master of Advanced Study from the University of\nCambridge, and a PhD from ETH Zürich. He has been awarded with two Gold\nmedals in Mathematical Olympiads in Greece, a Silver medal in the IMO,\nand with the Notara Prize of the Academy of Athens. He has published 16 books and volumes by Springer, including\n",
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" with J. F. Nash, Jr. He has published\nseveral research papers in Mathematical Analysis and Analytic Number\nTheory. His homepage is ",
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