Field Arithmetic

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Springer Science & Business Media, Apr 9, 2008 - Mathematics - 792 pages
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Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements.

Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)?

The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.

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Contents

I
1
IV
4
V
9
VI
12
VII
15
VIII
16
IX
18
X
19
CLI
350
CLII
358
CLIII
360
CLIV
361
CLV
362
CLVI
363
CLVII
366
CLVIII
370

XI
21
XII
24
XIII
30
XIV
34
XV
38
XVI
44
XVII
48
XVIII
50
XIX
51
XX
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XXI
54
XXII
56
XXIII
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XXIV
61
XXV
67
XXVI
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XXVII
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XXVIII
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XXIX
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XXX
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XXXI
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XXXII
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XXXIII
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XXXV
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XXXVI
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XXXVII
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XXXVIII
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XXXIX
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XL
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XLII
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XLIII
99
XLIV
104
XLV
105
XLVI
106
XLVII
107
XLVIII
111
XLIX
113
L
115
LI
121
LII
129
LIII
130
LIV
132
LV
134
LVI
135
LVII
137
LVIII
138
LIX
139
LX
141
LXI
145
LXII
147
LXIV
148
LXV
149
LXVII
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LXVIII
154
LXIX
156
LXX
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LXXI
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LXXII
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LXXIII
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LXXIV
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LXXV
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LXXVI
165
LXXVII
168
LXXVIII
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LXXIX
170
LXXXI
172
LXXXII
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LXXXIII
176
LXXXIV
178
LXXXV
180
LXXXVI
182
LXXXVII
185
LXXXVIII
187
LXXXIX
190
XC
191
XCI
192
XCIII
193
XCIV
199
XCV
201
XCVI
203
XCVII
207
XCVIII
211
XCIX
217
C
218
CI
219
CIII
223
CIV
224
CV
228
CVI
229
CVII
230
CVIII
231
CIX
236
CX
237
CXI
241
CXII
244
CXIII
248
CXIV
252
CXV
258
CXVI
262
CXVII
264
CXVIII
266
CXIX
267
CXX
268
CXXI
270
CXXII
272
CXXIII
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CXXIV
275
CXXV
276
CXXVI
277
CXXVII
279
CXXVIII
281
CXXIX
283
CXXX
286
CXXXI
289
CXXXII
290
CXXXIII
291
CXXXIV
294
CXXXV
298
CXXXVI
302
CXXXVII
306
CXXXVIII
308
CXXXIX
315
CXL
321
CXLI
325
CXLII
328
CXLIII
332
CXLIV
334
CXLV
335
CXLVI
336
CXLVII
338
CXLVIII
340
CXLIX
344
CL
346
CLIX
376
CLX
378
CLXI
380
CLXII
382
CLXIII
386
CLXIV
390
CLXV
394
CLXVI
397
CLXVII
400
CLXVIII
401
CLXIX
403
CLXX
406
CLXXI
411
CLXXII
412
CLXXIII
413
CLXXIV
422
CLXXV
425
CLXXVI
427
CLXXVII
428
CLXXVIII
429
CLXXXI
430
CLXXXII
433
CLXXXIII
436
CLXXXIV
438
CLXXXV
440
CLXXXVI
442
CLXXXVII
444
CLXXXVIII
446
CLXXXIX
448
CXC
451
CXCI
453
CXCII
454
CXCIII
455
CXCIV
460
CXCV
462
CXCVI
463
CXCVII
467
CXCVIII
472
CXCIX
479
CC
489
CCI
493
CCII
495
CCIII
497
CCIV
499
CCV
502
CCVI
506
CCVII
508
CCVIII
513
CCIX
515
CCX
520
CCXI
522
CCXII
524
CCXIII
528
CCXIV
532
CCXV
534
CCXVI
537
CCXVII
539
CCXVIII
542
CCXIX
544
CCXX
546
CCXXI
549
CCXXII
550
CCXXIII
554
CCXXIV
558
CCXXV
561
CCXXVII
562
CCXXVIII
565
CCXXIX
567
CCXXX
569
CCXXXI
574
CCXXXII
576
CCXXXIII
579
CCXXXIV
581
CCXXXV
583
CCXXXVI
586
CCXXXVII
591
CCXXXVIII
592
CCXXXIX
594
CCXL
595
CCXLI
601
CCXLII
604
CCXLIII
606
CCXLIV
613
CCXLV
615
CCXLVI
620
CCXLVII
623
CCXLVIII
625
CCXLIX
633
CCL
635
CCLII
640
CCLIII
642
CCLIV
646
CCLV
651
CCLVI
654
CCLVII
655
CCLIX
659
CCLX
664
CCLXI
667
CCLXII
670
CCLXIII
671
CCLXIV
672
CCLXV
676
CCLXVI
679
CCLXVII
682
CCLXVIII
687
CCLXIX
688
CCLXX
690
CCLXXI
694
CCLXXIII
697
CCLXXV
698
CCLXXVII
700
CCLXXVIII
704
CCLXXIX
706
CCLXXX
708
CCLXXXI
710
CCLXXXII
715
CCLXXXIII
717
CCLXXXIV
720
CCLXXXV
722
CCLXXXVI
725
CCLXXXVII
726
CCLXXXVIII
729
CCLXXXIX
730
CCXC
735
CCXCI
739
CCXCII
748
CCXCIII
750
CCXCIV
751
CCXCV
754
CCXCVI
758
CCXCVII
761
CCXCVIII
780
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About the author (2008)

Moshe Jarden (revised and considerably enlarged the book in 2004 (2nd edition) and revised again in 2007 (the present 3rd edition).


Born on 23 August, 1942 in Tel Aviv, Israel.

Education:
Ph.D. 1969 at the Hebrew University of Jerusalem on
"Rational Points of Algebraic Varieties over Large Algebraic Fields".
Thesis advisor: H. Furstenberg.
Habilitation at Heidelberg University, 1972, on
"Model Theory Methods in the Theory of Fields".

Positions:
Dozent, Heidelberg University, 1973-1974.
Seniour Lecturer, Tel Aviv University, 1974-1978
Associate Professor, Tel Aviv University, 1978-1982
Professor, Tel Aviv University, 1982-
Incumbent of the Cissie and Aaron Beare Chair,
Tel Aviv University. 1998-

Academic and Professional Awards
Fellowship of Alexander von Humboldt-Stiftung in Heidelberg, 1971-1973.
Fellowship of Minerva Foundation, 1982.
Chairman of the Israel Mathematical Society, 1986-1988.
Member of the Institute for Advanced Study, Princeton, 1983, 1988.
Editor of the Israel Journal of Mathematics, 1992-.
Landau Prize for the book "Field Arithmetic". 1987.
Director of the Minkowski Center for Geometry founded by the
Minerva Foundation, 1997-.
L. Meitner-A.v.Humboldt Research Prize, 2001
Member, Max-Planck Institut f\"ur Mathematik in Bonn, 2001.