Elements of analysis and algebra (and number theory) (Q2862340)

This comprehensive textbook on various fundamental topics in both classical and modern mathematics grew out of a course that the author has taught repeatedly for upper-undergraduate students at École Polytechnique in Paris, France. The first edition of this book was published in 2009, and a detailed review of it appeared right after the the issue of this fairly unique, highly remarkable volume [ibid. 469 p. (2009; Zbl 1188.00001)].NEWLINENEWLINE The book under review is the considerably expanded second edition of the original, where the nearly 200 new pages provide a wealth of additional preparatory material, diverse complements concerning the three advanced main topics as well as the numerous appendices of the text, and some new exercises (with solutions) from the final exams of the course.NEWLINENEWLINE As the original text of the book has been left largely intact, we may entirely refer to the review of the first edition [loc. cit.] with regard to both the structure and the contents of this volume, and confine ourselves here to expound some of the most important replenishments in the current second edition.NEWLINENEWLINE First of all, the preparatory chapter titled ``Mathematical Vocabulary'' has been amplified by about 100 pages. Having undergone a doubling in size the material of this introductory part has been rearranged so as to complete it by a significant number of additional basic concepts and results from group theory, linear algebra, general topology, real analysis, and the theory of number systems. As the author points out, this has been taken up in order to provide a rather complete version of the subject matter of the two-year preparatory classes for the study of mathematics at French universities.NEWLINENEWLINE Secondly, there is a new, additional appendix dealing primarily with the irrationality of infinitely many values of the Riemann zeta function. Further complements concern a complete construction of the \(p\)-adic zeta function (Appendix D), some remarks on Diophantine equations (Appendix F), and various other enrichments in the course of the entire text.NEWLINENEWLINE Finally, the concluding appendix presenting selected problems and their solutions which now appears as Appendix H, has been expanded by four additional sets of exercises taken from the final exams of the recent academic year. The topics discussed here are related to the determination of the characters of the simple group \(\text{GL}_3(\mathbb{F}_2)\), the Dedekind \(\eta\)-function, the irrationality of the zeta value \(\zeta(3)\), and a criterion or \textit{E. Borel} [C. R. Acad. Sci., Paris 120, 303--304 (1894; JFM 26.0429.02)], for the rationality of a complex power series with integer coefficients. As the author remarks in the preface, exercises of this particular type represent a special feature of the French school of higher mathematical education, as they permit to illustrate the fascinating unity of mathematics in a very instructure, albeit relatively elementary manner, and as they challenge students to combine effectively different mathematical concepts, methods, and results.NEWLINENEWLINE As for the present, enlarged second edition of this comprehensive textbook, it should emphasized again (and all the more) that this unique again treatise is of absolutely outstanding and original character, mainly in that it offers a tremendous wealth of fundamental mathematics at its best, with a wide panoramic view, numerous historical and cultural explanations, hundreds of carefully selected exercises, and all that in a very lucid, detailed, rigorous, enlightening and inspiring style of presentation. No doubt, the various complements and ameliorations make this fine textbook even more useful and valuable for students in the areas of mathematics, computer science physics, engineering, and mathematical education in high schools likewise.