Basic Algebra: Along with a Companion Volume Advanced Algebra, Volume 1

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Springer Science & Business Media, Sep 28, 2006 - Mathematics - 717 pages
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Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole.

Key topics and features of Basic Algebra:

*Linear algebra and group theory build on each other continually

*Chapters on modern algebra treat groups, rings, fields, modules, and Galois groups, with emphasis on methods of computation throughout

*Three prominent themes recur and blend together at times: the analogy between integers and polynomials in one variable over a field, the interplay between linear algebra and group theory, and the relationship between number theory and geometry

*Many examples and hundreds of problems are included, along with a separate 90-page section giving hints or complete solutions for most of the problems

*The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them; includes blocks of problems that introduce additional topics and applications for further study

*Applications to science and engineering (e.g., the fast Fourier transform, the theory of error-correcting codes, the use of the Jordan canonical form in solving linear systems of ordinary differential equations, and constructions of interest in mathematical physics) appear in sequences of problems

Basic Algebra presents the subject matter in a forward-looking way that takes into account its historical development. It is suitable as a text in a two-semester advanced undergraduate or first-year graduate sequence in algebra, possibly supplemented by some material from Advanced Algebra at the graduate level. It requires of the reader only familiarity with matrix algebra, an understanding of the geometry and reduction of linear equations, and an acquaintance with proofs.

  

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Contents

VII
1
VIII
4
IX
9
X
15
XI
19
XII
24
XIII
30
XIV
33
LX
292
LXI
303
LXII
314
LXIII
319
LXIV
326
LXV
342
LXVI
344
LXVII
367

XV
38
XVI
42
XVII
50
XVIII
54
XIX
58
XX
65
XXI
73
XXII
77
XXIII
82
XXIV
88
XXVI
98
XXVII
104
XXVIII
111
XXIX
116
XXX
117
XXXI
128
XXXII
134
XXXIII
140
XXXIV
147
XXXV
158
XXXVI
166
XXXVII
170
XXXVIII
174
XXXIX
183
XL
188
XLI
198
XLII
209
XLIV
212
XLV
216
XLVI
224
XLVII
226
XLVIII
229
XLIX
235
L
239
LI
245
LII
246
LIII
250
LIV
255
LV
257
LVI
260
LVII
274
LVIII
280
LIX
288
LXVIII
396
LXIX
408
LXX
414
LXXI
417
LXXII
425
LXXIII
434
LXXIV
448
LXXV
449
LXXVI
453
LXXVII
457
LXXVIII
460
LXXIX
464
LXXX
469
LXXXI
476
LXXXII
479
LXXXIII
483
LXXXIV
486
LXXXV
488
LXXXVI
493
LXXXVII
501
LXXXVIII
510
LXXXIX
514
XC
521
XCI
527
XCII
534
XCIII
544
XCIV
551
XCV
556
XCVI
558
XCVII
565
XCVIII
574
XCIX
579
C
583
CII
589
CIII
591
CIV
594
CV
595
CVI
603
CVII
697
CVIII
699
CIX
703
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