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E-grāmata: Emil Artin and Helmut Hasse: The Correspondence 1923-1958

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The exchange of letters, over more than three decades, between these two giants of mathematics affords a unique perspective on the evolution of class field theory and Artin’s reciprocity law. This is the first English translation of the correspondence.



This volume consists of the English translations of the letters exchanged between Emil Artin to Helmut Hasse written from 1921 until 1958. The letters are accompanied by extensive comments explaining the mathematical background and giving the information needed for understanding these letters. Most letters deal with class field theory and shed a light on the birth of one of its most profound results: Artin's reciprocity law.

Recenzijas

From the book reviews:

There is extensive mathematical and biographical commentary accompanying the letters, making it possible to follow the discussion without having to be familiar with class field theory and its history. For anyone interested in the history of number theory in the 20th century, this book is a must-have. (Fernando Q. Gouvźa, MAA Reviews, July, 2014)

Part I Introduction to the Correspondence
1(44)
1 Introduction
3(15)
2 Emil Artin, his life and his work
18(7)
3 Reminiscing Helmut Hasse
25(8)
4 Reciprocity Laws for Power Residues
33(5)
5 Class Field Theory
38(3)
6 Time Table
41(4)
Part II The Letters
45(414)
1 1923--1926
47(58)
1 09.07.1923, Letter from Artin to Hasse
48(3)
2 09.07.1923, Postcard from Artin to Hasse
51(1)
3 Date unknown, Letter from Artin to Hasse
52(4)
4 Date unknown, Letter from Artin to Hasse
56(3)
5 12.07.1923, Postcard from Artin to Hasse
59(22)
Comments on Letters 1-5
5.1 The Prehistory
62(1)
5.2 Hasse's Lecture in Hamburg
63(1)
5.3 Artin's Inversion Method
64(4)
5.4 Second Supplementary Law
68(4)
5.5 General Reciprocity Law
72(1)
5.6 The Article on X-Series
73(4)
5.7 The Gap 1923--1926
77(4)
6 10.02.1926, Letter from Artin to Hasse
81(20)
Comments
6.1 Complex Multiplication
88(3)
6.2 Topology
91(1)
6.3 Eisenstein's reciprocity law
92(1)
6.4 Cubic Gauss Sums
93(3)
6.5 Chebotarev's Article
96(5)
7 10.09.1926, Letter from Artin to Hasse
101(2)
Comments
7.1 Eisenstein's Method
103(2)
2 1927
105(114)
8 17.07.1927, Letter from Artin to Hasse
107(10)
Comments
8.1 Artin's Course in 1927
109(1)
8.2 The Jacobi Symbol
110(3)
8.3 Hasse's Plans for the Annalen
113(4)
9 19.07.1927, Letter from Artin to Hasse
117(18)
Comments
9.1 The Proof
125(1)
9.2 The Method of Abelian Crossing
125(4)
9.3 The Lemma
129(1)
9.4 Furtwangler's Trick
130(5)
10 21.07.1927, Letter from Artin to Hasse
135(14)
Comments
10.1 The σ-Formulation
138(1)
10.2 Extension of the Jacobi Symbol
139(4)
10.3 The Hilbert Symbol
143(3)
10.4 The Inversion Factor
146(1)
10.5 Part II of Hasse's Report on Class Field Theory
147(2)
11 26.07.1927, Letter from Artin to Hasse
149(9)
Comments
11.1 Primary and Hyperprimary Numbers
152(1)
11.2 On the Biquadratic Reciprocity Law
152(2)
11.3 Class Number Divisibility
154(4)
12 29.07.1927, Letter from Artin to Hasse
158(6)
Comments
12.1 The Structure of Class Field Theory
160(4)
13 02.08.1927, Letter from Artin to Hasse
164(11)
Comments
13.1 On the Principal Ideal Theorem
169(6)
14 06.08.1927, Letter from Artin to Hasse
175(9)
Comments
14.1 Relative Units
178(1)
14.2 Artin-Schreier and the Beautiful Formula
179(5)
15 19.08.1927, Letter from Artin to Hasse
184(17)
Comments
15.1 On the Class Field Tower
188(6)
15.2 Class Field Theory for Function Fields
194(2)
15.3 The Concept for Part II of Hasse's Report
196(3)
15.4 Non-abelian Extensions
199(2)
16 04.09.1927, Letter from Artin to Hasse
201(4)
Comments
16.1 The Second Supplementary Law
204(1)
17 27.10.1927, Letter from Artin to Hasse
205(14)
Comments
17.1 Reichenberg
211(1)
17.2 Joint Article on the Supplementary Laws
211(2)
17.3 Artin's Conjecture on Primitive Roots
213(3)
17.4 Explicit Formulas for the Reciprocity Law
216(1)
17.5 Minkowski and the Unit Theorem
217(2)
3 1928--1929
219(46)
18 05.08.1928, Letter from Artin to Hasse
221(5)
Comments
18.1 The Theorem of Jordan-Holder-Schreier
223(1)
18.2 The Hamburg Meeting 1928
223(1)
18.3 Hasse's Diary Entry
224(2)
19 04.11.1928, Letter from Artin to Hasse
226(3)
Comments
19.1 On the Density Theorem
228(1)
20 14.11.1928, Letter from Artin to Hasse
229(6)
Comments
20.1 Arnold Scholz
231(4)
21 18.11.1928, Letter from Artin to Hasse
235(4)
Comments
21.1 Furtwangler's Conjecture
236(3)
22 19.11.1928, Letter from Artin to Hasse
239(6)
Comments
22.1 Complex Quadratic Base Fields
241(1)
22.2 Ambiguous ideal classes
241(1)
22.3 Hasse's Generalization
242(1)
22.4 The group theoretic approach
242(1)
22.5 On Part II of Hasse's Report on Class Field Theory
243(2)
23 22.11.1928, Letter from Artin to Hasse
245(2)
24 03.12.1928, Letter from Artin to Hasse
247(2)
25 12.01.1929, Letter from Artin to Hasse
249(3)
Comments
25.1 Quadratic Forms
250(1)
25.2 Metabelian Groups
251(1)
26 10.05.1929, Letter from Artin to Hasse
252(7)
Comments
26.1 Hasse's Norm Residue Symbol
253(1)
26.2 Local Class Field theory
254(1)
26.3 Further Developments
255(1)
26.4 Ideal Theory
256(3)
27 19.05.1929, Letter from Artin to Hasse
259(3)
Comments
27.1 Petersson
260(2)
28 15.12.1929, Letter from Artin to Hasse
262(3)
Comments
28.1 Suetuna
263(1)
28.2 Continuous Groups
263(2)
4 1930--1934
265(146)
29 23.08.1930, Letter from Artin to Hasse
268(5)
Comments
29.1 The Proof Sheets
270(1)
29.2 Norm Residues
270(1)
29.3 The Inversion Factor
271(2)
30 18.09.1930, Letter from Artin to Hasse
273(15)
Comments
30.1 L-series
279(7)
30.2 In-primary
286(1)
30.3 Weddle
286(2)
31 23.09.1930, Letter from Artin to Hasse
288(11)
Comments
31.1 Two Publications
293(1)
31.2 Hasse's Article on the Conductor
294(1)
31.3 Book Projects
295(1)
31.4 Olga Taussky
296(3)
32 10.10.1930, Letter from Artin to Hasse
299(7)
Comments
32.1 Hasse's Communication
303(1)
32.2 The detour
304(2)
33 07.11.1930, Letter from Artin to Hasse
306(7)
Comments
33.1 The Conductor Manuscript
307(1)
33.2 Speiser
307(1)
33.3 Future Prospects
308(1)
33.4 Without Class Field Theory
309(4)
34 11.11.1930, Letter from Artin to Hasse
313(5)
Comments
34.1 The Remarks
315(1)
34.2 Zukunftsmusik
316(2)
35 27.11.1930, Letter from Artin to Hasse
318(7)
Comments
35.1 Corrections
320(1)
35.2 Hypercomplex Arithmetic
321(2)
35.3 Publication of Hasse's Report
323(2)
36 24.01.1931, Letter from Artin to Hasse
325(5)
Comments
36.1 Conjectures on Algebras
327(1)
36.2 The Gamma Function
328(1)
36.3 Schreier-Sperner
328(2)
37 06.05.1931, Letter from Artin to Hasse
330(4)
Comments
37.1 Icosahedral Extension
332(2)
38 16.06.1931, Letter from Artin to Hasse
334(15)
Comments
38.1 Maximal Orders
339(2)
38.2 Siegel
341(2)
38.3 Tremendous Simplifications
343(6)
39 24.08.1931, Letter from Artin to Hasse
349(7)
Comments
39.1 The Hensel Volume
350(1)
39.2 The Class Field Theory Proofs
351(1)
39.3 Herbrand
352(2)
39.4 The Theorem on Skew Fields
354(2)
40 November 1931, Letter from Artin to Hasse
356(6)
Comments
40.1 Cyclicity of Simple Algebras
358(1)
40.2 Artin's Amateurish Steps
359(3)
41 1932, Letter from Artin to Hasse
362(7)
Comments
41.1 Continuation of L-Series
364(1)
41.2 Algebras
365(4)
42 09.03.1932, Letter from Artin to Hasse
369(9)
Comments
42.1 First Letter on Factor Systems
374(4)
43 10.03.1932, Letter from Artin to Hasse
378(2)
44 11.03.1932, Postcard from Artin to Hasse
380(1)
45 ca. 16.03.1932, Letter from Artin to Hasse
381(6)
46 02.05.1932, Letter from Artin to Hasse
387(5)
Comments
46.1 Last Letter on Factor Systems
390(2)
47 08.11.1932, Letter from Hasse to Artin
392(10)
Comments
47.1 Artin's Class Field Theory Lectures
396(1)
47.2 The Inverse Theorem
397(5)
48 02.12.1932, Letter from Artin to Hasse
402(4)
Comments
48.1 The Proof of the Principal Ideal Theorem
404(1)
48.2 Hasse's Visit in Hamburg
405(1)
49 17.01.1934, Letter from Artin to Hasse
406(3)
Comments
49.1 Hasse's Work on the Riemann Conjecture in Function
Fields
407(2)
50 1934, Letter from Artin to Hasse
409(2)
5 1937--1938
411(12)
51 14.01.1937, Letter from Hasse to Artin
412(3)
51.1 Hasse's Visit
413(1)
51.2 Alice in Wonderland
414(1)
52 Nov. 1938, Natascha Artin to Hasse, undated [ Bloomington]
415(2)
53 09.12.1938, Letter from Hasse to Artin
417(6)
53.1 Hecke's Visit
419(1)
53.2 The Class Field Tower
419(1)
53.3 The Blue Hasse
420(3)
6 1953--1958
423(36)
54 12.03.1953, Letter from Artin to Hasse
425(4)
Comments
54.1 Artin's algebra seminar 1953/54
427(1)
54.2 Artin-Tate
427(2)
55 20.03.1953, Letter from Hasse to Artin
429(4)
Comments
55.1 Scholz's Number Knots
431(1)
55.2 Artin Root Numbers
431(2)
56 08.10.1955, Postcard from Artin to Hasse [ Tokyo]
433(3)
Comments
56.1 Tokyo-Nikko 1955
434(1)
56.2 Studium Generale
435(1)
57 20.10.1955, Postcard from Hasse to Artin
436(1)
58 20.02.1956, Letter from Artin to Hasse
437(1)
59 20.02.1956, Letter from Hasse to Artin
438(2)
59.1 Ernst Witt
439(1)
60 18.12.1956, Letter from Hasse to Artin
440(1)
61 04.01.1957, Letter from Artin to Hasse [ Gottingen]
441(1)
62 11.01.1957, Letter from Hasse to Artin
442(1)
63 15.11.1957, Letter from Artin to Hasse [ Princeton]
443(1)
64 28.11.1957, Letter from Hasse to Artin
444(1)
65 13.12.1957, Letter from Hasse to Artin
445(1)
66 15.12.1957, Letter from Artin to Hasse [ Princeton]
446(1)
67 20.01.1958, Letter from Artin to Hasse [ Princeton]
447(1)
68 11.03.1958, Letter from Artin to Hasse [ Princeton]
448(1)
69 20.03.1958, Letter from Hasse to Artin [ Ahrensburg]
449(2)
70 02.04.1958, Letter from Artin to Hasse [ Princeton]
451(1)
71 09.04.1958, Letter from Hasse to Artin
452(1)
72 17.04.1958, Letter from Artin to Hasse [ Princeton]
453(1)
73 23.04.1958, Letter from Hasse to Artin
454(1)
74 Report on Artin
455(4)
References 459(18)
Author Index 477(4)
Subject Index 481
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