The Birth of Algebraic Number Theory

Narkiewicz, Władysław

doi:10.1007/978-3-030-03754-3_1

Władysław Narkiewicz¹⁹

Part of the book series: Springer Monographs in Mathematics ((SMM))

Abstract

In the first chapter we present the development of the theory of algebraic numbers in the 19th century, describing concisely the work of Gauss, Dirichlet, Eisenstein, Kummer, Hermite, Kronecker and Dedekind.

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Notes

1.
Leonhard Euler (1765–1823), professor in St. Petersburg and Berlin. See [8, 1174, 4051, 4190].
2.
“Because if e.g., \(x^2+cy^2\) is a cube, then one can safely infer that its two irrational factors, namely \(x+y\sqrt{-c}\) and \(x-y\sqrt{-c}\), must be cubes, because they are co-prime as the numbers x and y do not have a common divisor.”
3.
Jean François Théophile Pépin (1826–1904), Jesuit, teacher of mathematics.
4.
Joseph Louis Lagrange (1736–1813), worked in Turin, Berlin and Paris. See [2082].
5.
Carl Friedrich Gauss (1777–1855), professor in Göttingen. See [1028, 2124].
6.
Ernst Eduard Kummer (1810–1893), professor in Breslau and Berlin. See [1782, 2410].
7.
“The whole theory of forms of second degree in two variables can be considered as the theory of complex numbers of the form \(x+y\sqrt{D}\).”
8.
Richard Dedekind (1831–1916), professor in Zürich and Braunschweig. See [836, 1718, 2429, 3107].
9.
Leopold Kronecker (1823–1891), professor in Berlin. See [1258, 2181, 4317].
10.
Andreas Speiser (1885–1970), professor in Zürich. See [1199].
11.
Robert König (1885–1979), professor in Tübingen, Münster, Jena and Munich.
12.
Salomon Lubelski (1902–?), worked in Warsaw, died in a Nazi concentration camp, date unknown.
13.
Irving Kaplansky (1917–2006), professor in Chicago and Berkeley. See [212].
14.
Hubert S. Butts (1923–1999), professor at the Louisiana State University.
15.
Dennis Ray Estes (1941–1999), professor at the University of Southern California.
16.
Martin Kneser (1928–2004), professor in München and Göttingen, son of Hellmuth Kneser. See [3943].
17.
The original definition of the composition is rather complicated. A simple way of defining it was provided in 1912 by Speiser [3869]
18.
Ernst Schering (1833–1897), professor in Göttingen. See [2168].
19.
Ferdinand Georg Frobenius (1849–1917), professor in Zürich and Berlin. See [1721].
20.
Ludwig Stickelberger (1850–1936), professor in Freiburg in Breisgau. See [1748].
21.
Robert Remak (1888–1942), dozent in Berlin.
22.
Alwin Reinhold Korselt (1864–1947), teacher in Plauen.
23.
Wolfgang Franz (1905–1996), professor in Frankfurt. See [517].
24.
Richard Rado (1906–1989), professor in Sheffield, Kings College London and Reading. See [2548, 3490].
25.
Eugene Schenkman (1922–1977), professor at the Purdue University.
26.
André Weil (1906–1998), professor in Sao Paulo, Chicago and at IAS in Princeton. See the special issue of Notices AMS, vol 46/4, 1999.
27.
Peter Gustav Lejeune-Dirichlet (1805–1859), professor in Breslau, Berlin and Göttingen. See [2879].
28.
Peter Friedrich Arndt (1817–1866), professor in Berlin.
29.
Franz Mertens (1840–1927), professor in Kraków, Graz and Vienna. See [930].
30.
Irving Reiner (1924–1986), professor at the University of Illinois in Urbana-Champaign. See [2035].
31.
Indar Singh Luthar (1932–2006), professor at the Panjab University.
32.
P. Charles Joubert (1825–1907), jesuit, teacher at Ste. Genévieve in Paris.
33.
Daniel Shanks (1917–1966), professor at the University of Maryland. See [4403].
34.
Edmund Landau (1877–1938), professor in Göttingen. See [1626, 2188].
35.
Matiaš Lerch (1860–1922), professor in Prague, Fribourg and Brno. See [801, 3336].
36.
Rudolf Lipschitz (1832–1903), professor in Breslau and Bonn. See [2225].
37.
Heinrich Weber (1842–1913), professor in Heidelberg, Zürich, Königsberg, Charlottenburg, Marburg, Göttingen and Strassburg. See [4249].
38.
Gordon Pall (1907–1987), professor at McGill University.
39.
Charles Hermite (1822–1901), professor in Paris. See [3128, 3283].
40.
Ivan Matveevič Vinogradov (1891–1983), professor in Moscow. See [601].
41.
Carl Ludwig Siegel (1896–1981), professor in Frankfurt and Göttingen. See [917, 1871, 3661].
42.
George Ballard Mathews (1861–1922), professor in Bangor and lecturer in Cambridge. See [325].
43.
Gotthold Eisenstein (1823–1852), dozent in Berlin. See [351, 4357].
44.
Pierre Wantzel (1814–1848), worked at L’École Polytechnique in Paris. See [552, 3554].
45.
Alfred Loewy (1873-1935), professor in Freiburg in Breisgau. See [1229].
46.
Julius Petersen (1839-1910), professor in Copenhagen. See [2672].
47.
James P. Pierpont (1866–1938), professor at Yale. See [3187].
48.
Mihály Bauer (1874–1945), professor in Budapest.
49.
Johann Gustav Hermes (1846–1912), teacher in Königsberg, Linden and Osnabrück.
50.
Leonard Eugene Dickson (1874–1945), professor in Chicago. See [52, 1178].
51.
Paul Lévy (1886–1971), professor in Paris. See [4039].
52.
Marc Krasner (1912–1985), professor in Clermont-Ferrand and at the Université Paris VI.
53.
Dmitriĭ Abramovič Raĭkov (1905–1980), professor in Moscow.
54.
John Knopfmacher (1937–1999), father of Arnold Knopfmacher, professor at the University of the Witwatersrand. See [3710].
55.
Edmund Busche (1861–1916), teacher in Bergedorf and Hamburg. See [3463].
56.
David Hilbert (1862–1943), professor in Königsberg and Göttingen. See [3420, 3425, 4379].
57.
Heinrich Dörrie (1873–1955), teacher in Biedenkopf and Wiesbaden.
58.
Erwin Dintzl (1878–1972).
59.
Gottfried Martin Rückle (1879–1929).
60.
Erich Hecke (1887–1947), professor in Göttingen and Hamburg. See [3266, 3662].
61.
Vladimir Petrovič Vel’min (1885–1974), professor in Warsaw, Rostov and Kiev. See [2094].
62.
Carl Gustav Jacob Jacobi (1804–1851), professor in Königsberg and Berlin. See [2210, 2211].
63.
Walther Lietzmann (1880–1959), teacher in Barmen, Jena and Göttingen.
64.
Stjepan Bohniček (1872–1956), professor in Zagreb.
65.
Émile Mathieu (1835–1890), professor in Besançon and Nancy. See [1018].
66.
“...I do not believe that only arithmetics led to such a hidden idea, but that it was drawn from a study of elliptical transcendentals, from a special kind of them giving the rectification of the arc of the lemniscate.”
67.
Niels Henrik Abel (1802–1829), docent in Christiania (Oslo). See [2486, 3188, 3942].
68.
Giulio Carlo Fagnano dei Toschi (1682–1766). See [160].
69.
Joseph Liouville (1809–1882), professor in Paris. [2670].
70.
Ludwig Kiepert (1841–1934), professor in Hannover.
71.
Karl Schwering (1846–1925), teacher in Köln.
72.
“The theory of cubic residues should be similarly based on numbers of \(a+bh\), where h is a certain complex root of the equation \(h^3-1=0\), e.g., \(h=-\frac{1}{2}+\sqrt{\frac{3}{4}}i\), and in exactly the same way the theory of residues for higher exponents needs the introduction of other imaginary quantities.”
73.
Adrien-Marie Legendre (1752–1833), professor in Paris.
74.
Gabriel Lamé (1795–1870), professor in St. Petersburg and at École Polytechnique in Paris. See [4242].
75.
Augustin Louis Cauchy (1789–1851), professor in Paris. See [280, 4123].
76.
Victor Amédée Lebesgue (1791–1875), professor in Bordeaux.
77.
“One could consider instead of expressions of the form \(t+u\sqrt{-1}\) also expressions of the form \(t+u\sqrt{a}\), a being without square factors. Expressions of this kind considered from the same point of view satisfy theorems analogous to those which are subject of this memoire and possibly can be proved in a similar way.”
78.
Already in 1863 Schering found a proof but it was published only in 1909 [3601].
79.
Arnold Meyer (1844–1896), professor in Zürich. See [2451].
80.
Charles de la Vallée-Poussin (1866–1962), professor in Louvain. See [519].
81.
Paul Isaac Bernays (1888–1977), professor in Göttingen, worked later at ETH in Zürich.
82.
Hartmut Ehlich (1931–2001), professor in Bochum.
83.
Paul Bachmann (1837–1920), professor in Breslau and Münster. See [1797].
84.
Luigi Bianchi (1856–1928), professor in Pisa. See [1287].
85.
“It seems difficult to establish this by purely arithmetic considerations.”
86.
Gustav Herglotz (1881–1953), professor in Leipzig. See [4064].
87.
Sigekatu Kuroda (1905–1972), professor in Nagoya and at the University of Maryland. See [2556].
88.
Hans Reichardt (1908–1991), professor in Berlin. See [2197].
89.
It seems that the first mention of units in a non-quadratic field occurs in the letter of Jacobi to Legendre of 27 May 1832 ([2], 275–277 and [3287], 80–83) in which he announces a result equivalent to the existence of infinitely many units in biquadratic fields.
90.
The complete proof appeared in his lectures, see [972], §166 in the second edition.
91.
Jules Molk (1857–1914), professor in Besançon and Nancy. See [4236].
92.
Bartel Leendert van der Waerden (1903–1996), professor in Leipzig, Amsterdam and Zürich. See [1240, 3721]
93.
Jacobi published his proof in [2010] in 1846.
94.
It is remarkable that in 1844 Eisenstein published 22 papers and three lists of problems in the Journal für reine und angewandte Mathematik.
95.
“Although elementary theorems of the theory of integral complex numbers of the form \(a+b\varrho \), where \(\varrho \) denotes an imaginary cubic root of unity are to be found nowhere, we believe that in view of the great analogy between these complex numbers and the usually called complex integers of the form \(a+b\sqrt{-1}\), we may assume the knowledge of these theorems as far as they deal with divisibility, factorization into simpler factors, the theory complex primes, etc.”
96.
Victor von Dantscher (1847–1921), professor in Graz.
97.
Leopold Gegenbauer (1807–1894), professor in Czernowitz, Innsbruck and Vienna. See [2189].
98.
Lothar Koschmieder (1890–1974), professor in Brno, Graz, Aleppo, Tucumán, Baghdad and Tübingen.
99.
Josef Anton Gmeiner (1862–1927), professor in Prague and Innsbruck.
100.
Helmut Hasse (1898–1979), professor in Halle, Marburg, Göttingen, Berlin and Hamburg. See [1238].
101.
Moritz Abraham Stern (1807–1894), professor in Göttingen. See [3529].
102.
“Also Jacobi supports my view that the theory of general complex numbers can be completed only by the complete theory of higher forms”
103.
Clairborne Green Latimer (1893–1960), professor at Tulane, Kentucky University and Emory.
104.
Lemmermeyer wrote on p. 311 of his book [2527]: “Given that his paper appeared in a rather obscure journal, it is surprising that it has been noticed at all”.
105.
Later such integers were called semi-primary (see [1836], §115.)
106.
Philipp Furtwängler (1869–1940), professor in Bonn, Aachen and Vienna. See [1885, 1915].
107.
Alfred Edward Western (1873–1961), worked as a solicitor. See [2863].
108.
Rudolf Fueter (1880–1950), professor in Zürich. See [3874].
109.
Carl Gustav Reuschle (1812–1875), teacher in Tübingen and Stuttgart. See [3106].
110.
Henry William Lloyd Tanner (1851–1915), professor in Cardiff.
111.
Nowadays the name “cyclotomic unit” is used for the elements of the group generated by \(\pm \zeta _p\) and \(1-\zeta _p^a\) (\(a=1,2,\dots , p-1\)).
112.
Jacques Herbrand (1908–1931). See [943].
113.
“the theorem yet to be proved”.
114.
“one of my main results, on which I built since a quarter of the year, asserting that the second factor of the class-number \(\frac{D}{\varDelta }\) is never divisible by \(\lambda \), is false, or at least unproved”.
115.
Harry Schultz Vandiver (1882–1973), professor at the University of Texas. See [772].
116.
For expositions of Kummer’s work on Fermat’s theorem see Bachmann [173] and Edwards [1043–1045]
117.
Kummer actually used another numeration of Bernoulli numbers, omitting the vanishing terms with odd indices \(>1\)
118.
Kurt Hensel (1861–1941), professor in Berlin and Marburg, edited the Journal für reine und angewandte Mathematik from 1901 on. See [1693].
119.
Dmitry Mirimanoff (1861–1945), professor in Geneva. See [4182].
120.
Kuusta Adolf Inkeri (1908–1997), professor in Turku.
121.
Leonard Carlitz (1907–1999), professor at the Duke University. See [437].
122.
Derrick Henry Lehmer (1907–1999), son of Derrick Norman Lehmer, husband of Emma Lehmer, professor at Berkeley. See [454].
123.
Emma Lehmer (1906–2001), wife of Derrick Henry Lehmer. See [455].
124.
Standard Western Automatic Computer, constructed in 1950, with 9472-bit memory. See Corry [773].
125.
John Lewis Selfridge (1927–2010), professor at the University of Illinois and the Northern Illinois University.
126.
Richard Crandall (1947–2012), professor at Reed College in Portland.
127.
By Generalized Riemann Hypothesis we understand the assertion that Dedekind zeta-functions do not have zeros in the half-plane \(\mathfrak {R}s>1/2\).
128.
Albrecht Fröhlich (1916-2001), professor at the King’s College, London. See [362, 4036].
129.
Richard Anthony Mollin (1947–2014), professor in Calgary.
130.
Up to now (2017) this has been done only for \(p<151\) (J.C. Miller [2862]).
131.
Paul Wolfskehl (1856–1906), lectured in Darmstadt. See [191].
132.
Hermann Minkowski (1864–1909) professor in Bonn, Königsberg, Zürich and Göttingen. See [1844].
133.
Felix Bernstein (1878–1956), professor in Göttingen, New York and Syracuse. See [1246].
134.
Max Gut (1898–1988), professor in Zürich.
135.
Kenkichi Iwasawa (1917–1998), professor in Tokyo, at MIT and in Princeton. [722].
136.
Lazarus Fuchs (1833–1902), professor in Berlin, Greifswald, Göttingen and Heidelberg. See [1613].
137.
If n is an arbitrary integer, and \(\omega \) is a primitive n-th root of unity, then the second factor is by itself the class-number of the complex numbers constructed from the numbers \(\omega +\omega ^{-1}, \omega ^2+\omega ^{-2}, \omega ^3+\omega ^{-3}, \dots \)”.
138.
...about the ideal prime factors of complex numbers, built of roots of an arbitrary irreducible equation of nth degree I do not have any clear idea.
139.
These theorems, which were given by Gauss for quadratic forms hold also for all systems of non-equivalent ideal numbers, which one can construct from complex numbers built from roots of arbitrary algebraic equations, not only from roots of the equation \(\alpha ^\lambda =1\).
140.
Nicolaas George Wijnand Henri Beeger (1884-1965), teacher.
141.
Eduard Selling (1834–1920), professor in Würzburg.
142.
Joseph Alfred Serret (1819–1885), professor in Paris. See [2074].
143.
Karl Grandjot (1900–1979), professor at the University of Chile. See [1563].
144.
Issai Schur (1875–1941), professor in Bonn and Berlin. See [2079].
145.
Friedrich Wilhelm Levi (1888–1966), professor in Leipzig, Calcutta, Bombay, and the Freie Universität Berlin. See [1295].
146.
Thoralf Skolem (1887–1963), professor in Oslo and Bergen. [3033].
147.
Josip Plemelj (1873-1967), professor at the TU Wien, in Czernowitz and Ljubljana.
148.
Angelo Genocchi (1817–1889), professor in Torino. See [3232].
149.
Trygve Nagell (1895-1988), professor in Uppsala. See [597].
150.
Hans Petersson (1902–1984), professor in Hamburg and Münster. See [4434].
151.
Tom Mike Apostol (1923–2016), professor at Caltech.
152.
Paul Erdős (1913–1996), professor in Budapest. See [164, 165, 1594].
153.
Note that at that time Kronecker used the word “Abelian” to mean “cyclic”. Cf. p. 237 of the paper by Petri and Schappacher [3274].
154.
Boris Nikolajevič Delone (Delaunay) (1890–1980), professor in Leningrad and Moscow.
155.
Igor Rostislavovič Šafarevič (1923–2017), professor in Moscow.
156.
Nikolaĭ Grigorievič Čebotarev (1894–1947), professor in Kazan. In publications in German language he spelled his name Tschebotareff and Tschebotaröw. See [881].
157.
Karel Petr (1868–1950), professor in Prague. See [2233].
158.
Adolf Hurwitz (1859–1919), brother of Julius Hurwitz, professor in Königsberg and Zürich. See [1845, 3203, 4457].
159.
Nicolas I Bernoulli (1687–1759), professor in Padua and Basel.
160.
Friedrich Theodor von Schubert (1758–1825), great grandfather of Sofija Kovalevskaja, worked in St. Petersburg.
161.
Carl Runge (1856–1927), professor in Hannover and Göttingen. See [785].
162.
Max Mandl (1859–1910), teacher in Prostějov and Lublana.
163.
Harris Hancock (1867–1944), professor in Chicago and Cincinnati. See [2922].
164.
Morgan Ward (1901–1963), professor at Caltech. See [2514].
165.
See the report on Weber’s book by Pierpont [3291].
166.
“The variables used in the theory of algebraic numbers do not have the meaning of variable sequences of numbers, ..., but are only symbols for calculation, without having any independent meaning.”
167.
Julius [Gyula] Kőnig (1849–1913), professor in Budapest.
168.
After defining the discriminant Hermite wrote on p. 335: “...laquelle les géométres anglais ont donné le nom de discriminant.” (“which the English geometers have given the name discriminant”).
169.
Julius Hurwitz (1857–1919), brother of Adolf Hurwitz, Privatdozent in Basel. See [3203].
170.
Lester Randolph Ford (1886–1967), professor at the Illinois Institute of Technology.
171.
Oskar Perron (1880–1975), professor in Tübingen, Heidelberg and Munich. See [1230, 1758].
172.
Nikolaus Hofreiter (1904–1990), professor in Vienna. See [1872].
173.
Georges Poitou (1926–1989), professor in Lille and at the Université Paris-Sud.
174.
Josef Wellstein (1869–1919), professor in Giessen and Strassburg.
175.
Chaim Herman Müntz (1884–1956). See [3194].
176.
Louis Joel Mordell (1888–1972), professor in Manchester and Cambridge. See [596, 825].
177.
Hans Zassenhaus (1912–1991), professor in Hamburg, at the McGill University, the University of Notre Dame and the Ohio State University. See [3318].
178.
Hans Frederik Blichfeldt (1873–1945), professor at Stanford University. See [940].
179.
Arnold Scholz (1904–1942), professor in Kiel. See [4024].
180.
Claude Ambrose Rogers (1920–2005), professor in Birmingham and at the University College, London.
181.
Anne-Marie Bergé (1939–2008), professor in Bordeaux. See [2756].
182.
Felix Klein (1849–1925), professor in Erlangen, München, Leipzig and Göttingen. See [784].
183.
Hans Rohrbach (1903–1993), professor in Mainz.
184.
Ralph Tambs-Lyche (1890–1991), professor in Oslo.
185.
Emil Artin (1898--1962), professor in Hamburg, Princeton and at Notre Dame University and Indiana University. See [685, 1026].
186.
Boris Faddeevič Skubenko (1929–1993). See [81].
187.
X Supplement in the second edition and XI Supplement in the next two.
188.
In a paper published in 1882 jointly with H. Weber [857] Dedekind extended this theory to fields of algebraic functions.
189.
Öystein Ore (1899–1968), professor at Yale. See [11].
190.
This proof is included in §172 of [844] in Dedekind’s collected papers [856].
191.
Dedekind called them “einartige Ideale”.
192.
In [856] the editors inserted into the text of [844] a proof of this assertion taken from Dedekind’s manuscripts.
193.
Adolf Kneser (1862–1930), professor in Dorpat, Berlin and Breslau, father of Hellmuth Kneser. See [2228].
194.
Georg Landsberg (1865–1912), professor in Heidelberg, Breslau and Kiel.
195.
Already in [842] he mentioned the existence of such condition, pointing out that its determination is not particularly difficult.
196.
Heinrich Grell (1903–1974), professor in Berlin. See [5].
197.
George David Birkhoff (1884–1944), professor at Princeton and Harvard. See [4394].
198.
Theodor Schönemann (1812–1868), teacher in Brandenburg.
199.
Leo Königsberger (1837–1921), professor in Greifswald, Heidelberg, Dresden and Vienna. See [394].
200.
Eugen Netto (1848–1918), professor in Strassburg, Berlin and Giessen. See [1693].
201.
Saunders Mac Lane (1909–2005), professor at Harvard and in Chicago. See [2690].
202.
See also [846].
203.
Lemmermeyer notes in his book [2527] (p. 125) that a part of its fame “is due to its name in German, where it is often called the Führerdiskriminantenproduktformel”.
204.
Alfredo Capelli (1855–1910), professor in Palermo and Naples. See [4069].
205.
Ernst Adolf Wendt (1872–1946).
206.
Karl Theodor Vahlen (1869–1945), professor in Greifswald and Berlin. See [3799].
207.
Jacob Westlund (1868–1947, professor at Purdue University.
208.
William Edward Hodgson Berwick (1888–1944), professor in Bangor. See [819].
209.
Udo Wegner (1902–1989), professor in Darmstadt, Heidelberg, Saarbrücken and Stuttgart.
210.
Heinrich Franz Friedrich Tietze (1880–1964), professor in Erlangen and Munich. See [3260].
211.
Henry Berthold Mann (1905–2000), professor at the Ohio State University, Univ. of Wisconsin and Univ. of Arizona. See [3152].
212.
“Dirichlet’s ideal function”
213.
Georgij Rabinowitsch = Georg Yuri Rainich (1886–1968), professor at the University of Michigan.
214.
Actually the proof of this result was found by Dedekind, who communicated it to Frobenius in June 1882. See [847].
215.
Georgiĭ Fedoseevič Voronoĭ (1868–1908), professor in Warsaw. See [3925].
216.
Emanuel Lasker (1868–1941), world chess champion in 1894–1921.
217.
Aleksandr Aleksandrovič Friedmann (1888–1925), professor in Perm. See [1183].
218.
Jacob David [Yacov Davidovič] Tamarkin (1888–1945), professor in Perm, St. Petersburg and at the Brown University. See [1851].
219.
James Whitbread Lee Glaisher (1848–1928), worked in Trinity College, Cambridge.
220.
Michel Plancherel (1885–1967), professor in Fribourg and Zürich.
221.
Egor Ivanovič Zolotarev (spelled also Zolotareff) (1847–1878), professor in St. Petersburg. See [2390, 3206].
222.
Julian Sochocki (1842–1927), professor in St. Petersburg.
223.
Karel Rychlik (1885–1968), professor in Prague. See [1943].
224.
Andrei Andreyevič Markov (1856–1922), professor in St. Petersburg. See [1523, 3917].
225.
Ivan Ivanovič Ivanov (1862–1939), professor in St. Petersburg. See [2389].
226.
Dmitriĭ Aleksandrovič Grave (1863–1939), professor in Kharkov and Kiev. See [880, 977].
227.
Julius Sommer (1871–1943), professor at the Technische Hochschule in Danzig.
228.
Harald Bergström (1908–2001), professor in Göteborg.
229.
Dmitriĭ Konstantinovič Faddeev (1907–1989), professor in Leningrad. See [57, 3553].
230.
Henri Poincaré (1854–1912), professor in Paris. See [3284] and the volume 38 (1921) of Acta Mathematica.
231.
Auguste Bravais (1811–1863), professor in Lyon and Paris.
232.
Alexander Axer (1880–1948), teacher in Zürich.
233.
Alexander v.Brill (1842–1935), professor in Darmstadt, München and Tübingen. See [1187, 2622].
234.
Masayoshi Nagata (1927–2008), professor in Kyoto. See [2898].
235.
Lenstra wrote in [2544]: “His proof is probably not correct, but its sketchiness makes this difficult to confirm.”
236.
James Victor Uspensky [Jakov Viktorovič Uspenskiĭ] (1883–1947), professor in St. Petersburg and at Stanford University.
237.
Tito Chella (1881–1923), professor in Pisa.
238.
Herbert James Godwin (1916–2009), professor at the Royal Holloway College, London.
239.
Ekaterina Alekseevna Naryškina (1895–1940).
240.
Henry John Stephen Smith (1826–1883), professor in Oxford.
241.
In his survey on number theory, published in six parts in the Report of the British Association for the years 1859–1865 Smith [3833] presented also the first steps of the theory of algebraic numbers [3832]
242.
Arthur Cayley (1821–1895), professor in Cambridge. See [3127].
243.
William Burnside (1852–1927), professor in Greenwich. See [1219].
244.
Arthur Stafford Hathaway (1855–1934), professor at the Cornell University and in Terre Haute.
245.
George Abram Miller (1863–1951), professor at the Cornell University, at Stanford and the University of Illinois at Urbana-Champaign. See [422].

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University of Wrocław, Wrocław, Poland
Władysław Narkiewicz

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Władysław Narkiewicz
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Narkiewicz, W. (2018). The Birth of Algebraic Number Theory. In: The Story of Algebraic Numbers in the First Half of the 20th Century. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-03754-3_1

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DOI: https://doi.org/10.1007/978-3-030-03754-3_1
Published: 19 January 2019
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