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Chebotarëv and his density theorem. (English) Zbl 0885.11005
The authors sketch the life of N. G. Chebotarëv (1894-1947), most famous for his density theorem in algebraic number theory. Two-thirds of the paper are devoted to Chebotarëv’s mathematics. These include a brief history and explanation of the density theorem and its importance, together with a sketch of a proof of it which does not use class field theory (the historical situation). Also included are two further results of Chebotarëv; one on Vandermonde determinants, which has application to the singularities of gap series, and one dealing with whether a certain class of numbers is constructible.
The paper, both mathematically and historically, is very clearly written. There is a picture of Chebotarëv. One small note: while transliteration is always a problem, and the Mathematical Review Journals’ standard is an appropriate one to follow, it is nevertheless a little strange to see Alexander Ostrowski (who, though Russian, spent his career in German speaking institutions, chiefly Göttingen and Basel) referred to as Ostrovskii.

MSC:
11-03 History of number theory
11R45 Density theorems
01A60 History of mathematics in the 20th century
01A70 Biographies, obituaries, personalia, bibliographies
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[1] E. Artin, Über eine neue Art von L-Reihen, Abh. Math. Sem. Univ. Hamburg 3 (1923), 89–108; Collected papers, pp. 105–124. Addison-Wesley, Reading, MA, 1965. · JFM 49.0123.01 · doi:10.1007/BF02954618
[2] E. Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. Math. Sem. Univ. Hamburg 5 (1927), 353–363; Collected papers, pp. 131–141. Addison-Wesley, Reading, MA, 1965. · JFM 53.0144.04 · doi:10.1007/BF02952531
[3] G. N. Chebotarëv,Iz vospominaniî ob ottse (Front the recollections on my father) (unpublished).
[4] N. G. Chebotarëv, Opredelenie plotnosti sovokupnosti prostykh chisel, prinadlezhashchikh zadannomu klassu podstanovok (Determination of the density of the set of prime numbers, belonging to a given substitution class), Izv. Ross. Akad. Nauk 17 (1923), 205–250; Sobranye sochineniî I, 27-65.
[5] N. Tschebotareff (= N.G. Chebotarëv), Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören, Math. Ann. 95 (1925), 191–228. · JFM 51.0149.04 · doi:10.1007/BF01206606
[6] N. Tschebotaröw (= N.G. Chebotarëv), Über quadrierbare Kreisbogenzweiecke, I, Math. Z. 39 (1935), 161–175. · JFM 60.0069.01 · doi:10.1007/BF01201352
[7] N. G. Chebotarëv,Sobranye sochinenî (Collected works), Akademiya Nauk SSSR, Moscow, 1949–1950 (3 volumes).
[8] N. G. Chebotarëv, Letter to Mikhail Il’ich Rokotovskiî, July 3, 1945, in Pis’ma i Vospominaniya (Letters and Recollections) (16 pp., unpublished).
[9] N. Tschebotaröw (= N.G. Chebotarëv),Grundzüge der Galois’sehen Theorie, übersetzt und bearbeitet von H. Schwerdtfeger, Noordhoff, Groningen, 1950.
[10] Th. Clausen, Vier neue mondförmige Flächen, deren Inhalt quadrirbar ist, J. Reine Angew. Math. 21 (1840), 375–376. · ERAM 021.0673cj · doi:10.1515/crll.1840.21.375
[11] Ch. de la Vallée-Poussin, Recherches analytiques sur la théorie des nombres premiers. Deuxième partie: Les fonctions de Dirichlet et les nombres premiers de la forme linéaireMx + N, Ann. Soc. Sci. Bruxelles 20 (1896), 281–362.
[12] M. Deuring, Über den Tschebotareffschen Dichtigkeitssatz, Math. Ann. 110 (1935), 414–415. · Zbl 0009.39403 · doi:10.1007/BF01448036
[13] J. Dieudonné, Une propriété des racines d’unité, Rev. Un. Mat. Argentina 25 (1970), 1–3; Math. Rev. 47, #8495; see also [7], Vol. 3, p. 162; Math. Rev. 17, 338x; Math. Rev. 53, #7997.
[14] G. Lejeune Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Abh. Königl. Akad. Wissenschaft. Berlin, math. Abh. (1837), 45–71; Werke I, pp. 313-342. Georg Reimer, Berlin, 1889.
[15] A. V. Dorodnov, O krugovykh lunochkakh, kvadriruemykh pri pomoshchi tsirkulya i lineîki (On circular lunes quadrable with the use of ruler and compass), Dokl. Akad. Nauk SSSR (N.S.) 58 (1947), 965–968.
[16] G. Frei,Die Briefe von E. Artin an H. Hasse (1923–1953), Collection Mathématique, Département de Mathématiques, Université Laval, Québec, 1981.
[17] F. G. Frobenius, Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe, Sitzungsberichte Königl. Preußisch. Akad. Wissenschaft. Berlin (1896), 689–703; Gesammelte Abhandlungen II, 719–733. Springer, Berlin, 1968. · JFM 27.0091.04
[18] H. Hasse, History of class field theory,Algebraic Number Theory, Proceedings of an Instructional Conference, (J. W. S. Cassels and A. Fröhlich, eds), Academic Press, London, 1967, pp. 266–279.
[19] T. Heath,A History of Greek Mathematics, Oxford University Press, Oxford, 1921, Vol. I. · JFM 48.0046.01
[20] E. Hecke, Über die L-Funktionen und den Dirichletschen Primzahlsatz für einen beliebigen Zahlkörper, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. (1917), 299–318; Mathematische Werke, 178–197. Vandenhoeck & Ruprecht, Göttingen, 1959. · JFM 46.0256.03
[21] Istoriya otechestvennoî matematiki (History of our national mathematics), Naukovo Dumka, Kiev, 1969, vol. 3.
[22] E. R. Kolchin, Math. Rev. 17 (1956), 1045.
[23] L. Kronecker, Über die Irreductibilität von Gleichungen, Monatsberichte Königl. Preußisch. Akad. Wissenschaft. Berlin (1880), 155–162; Werke II, 83–93. B. G. Teubner, Leipzig, 1897. · JFM 12.0065.02
[24] S. Lang,Algebraic Number Theory, Addison-Wesley, Reading, MA, 1970. · Zbl 0211.38404
[25] R. Lidl and H. Niederreiter,Finite Fields, Addison-Wesley, Reading, MA, 1983.
[26] Matematika v SSSR za 30 let, 1917–1947 (Mathematics in the USSR after 30 years, 1917–1947), OGIZ, Moscow, 1948.
[27] V. V. Morozov, Nikolaî Grigor’evich Chebotarëv (28 pp., unpublished).
[28] V. V. Morozov, Kazanskaya matematicheskaya shkola za 30 let – algebra (The Kazan mathematical school after 30 years – algebra), Usp. Mat. Nauk 2(6) (1947), 3–8.
[29] N. G. Chebotarëv– nekrolog (N. G. Chebotarëv – obituary), Usp. Mat. Nauk 2(6) (1947), 68–71.
[30] J. Neukirch,Class Field Theory, Springer-Verlag, Berlin, 1986. · Zbl 0587.12001
[31] R. W. K. Odoni, A conjecture of Krishnamurthy on decimal periods and some allied problems, J. Number Theory 13 (1981), 303–319. · Zbl 0471.10031 · doi:10.1016/0022-314X(81)90016-0
[32] A. M. Ostrowski, Über Singularitäten gewisser mit Lücken behafteten Potenzreihen. Mathematische Miszellen, VII, Jahresber. Deutsch. Math.-Verein. 35 (1926), 269–280; Collected mathematical papers 5, 181–192. Birkhäuser, Basel, 1985. · JFM 52.0293.01
[33] O. Schreier, Über eine Arbeit von Herrn Tschebotareff, Abh. Math. Sem. Univ. Hamburg 5 (1927), 1–6. · JFM 52.0166.02 · doi:10.1007/BF02952505
[34] J-P. Serre,Abelian l-Adic Representations and Elliptic Curves, W. A. Benjamin, New York, 1969.
[35] J-P. Serre, Quelques applications du théorème de densité de Chebotarëv, Publ. Math. I.H.E.S. 54 (1981), 123–201; OEuvres III, 563–641. Springer, Berlin, 1986.
[36] A. L. Shields, Luzin and Egorov, Math. Intelligencer 9(4) (1987), 24–27. · Zbl 0632.01022 · doi:10.1007/BF03023726
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