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E-raamat: Algebra 1: Groups, Rings, Fields and Arithmetic

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Algebra 1: Groups, Rings, Fields and ArithmeticSuurem pilt
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This is the first in a series of three volumes dealing with important topics in algebra. It offers an introduction to the foundations of mathematics together with the fundamental algebraic structures, namely groups, rings, fields, and arithmetic. Intended as a text for undergraduate and graduate students of mathematics, it discusses all major topics in algebra with numerous motivating illustrations and exercises to enable readers to acquire a good understanding of the basic algebraic structures, which they can then use to find the exact or the most realistic solutions to their problems.

Arvustused

The rich material is explaind in a very lucid, detailed and rigorous style. Together with the unique wealth of instructive illustrations, examples, and exercises, the main text makes an excellent source for both teaching and self-study of the subject. Certainly, this new textbook is a very useful and valuable enrichment of the plenty of existing primers in the field of higher algebra. (Werner Kleinert, zbMATH 1369.00002, 2017)

1 Language of Mathematics 1 (Logic)
1(12)
1.1 Statements, Prepositional Connectives
1(2)
1.2 Statement Formula and Truth Functional Rules
3(4)
1.3 Quantifiers
7(1)
1.4 Tautology and Logical Equivalences
8(1)
1.5 Theory of Logical Inference
9(4)
2 Language of Mathematics 2 (Set Theory)
13(42)
2.1 Set, Zermelo--Fraenkel Axiomatic System
13(9)
2.2 Cartesian Product and Relations
22(4)
2.3 Equivalence Relation
26(3)
2.4 Functions
29(9)
2.5 Partial Order
38(5)
2.6 Ordinal Numbers
43(5)
2.7 Cardinal Numbers
48(7)
3 Number System
55(38)
3.1 Natural Numbers
55(4)
3.2 Ordering in N
59(3)
3.3 Integers
62(9)
3.4 Greatest Common Divisor, Least Common Multiple
71(8)
3.5 Linear Congruence, Residue Classes
79(7)
3.6 Rational Numbers
86(2)
3.7 Real Numbers
88(3)
3.8 Complex Numbers
91(2)
4 Group Theory
93(52)
4.1 Definition and Examples
94(12)
4.2 Properties of Groups
106(7)
4.3 Homomorphisms and Isomorphisms
113(9)
4.4 Generation of Groups
122(12)
4.5 Cyclic Groups
134(11)
5 Fundamental Theorems
145(34)
5.1 Coset Decomposition, Lagrange Theorem
145(10)
5.2 Product of Groups and Quotient Groups
155(18)
5.3 Fundamental Theorem of Homomorphism
173(6)
6 Permutation Groups and Classical Groups
179(40)
6.1 Permutation Groups
179(8)
6.2 Alternating Maps and Alternating Groups
187(12)
6.3 General Linear Groups
199(10)
6.4 Classical Groups
209(10)
7 Elementary Theory of Rings and Fields
219(50)
7.1 Definition and Examples
219(2)
7.2 Properties of Rings
221(3)
7.3 Integral Domain, Division Ring, and Fields
224(9)
7.4 Homomorphisms and Isomorphisms
233(5)
7.5 Subrings, Ideals, and Isomorphism Theorems
238(12)
7.6 Polynomial Ring
250(11)
7.7 Polynomial Ring in Several Variable
261(8)
8 Number Theory 2
269(42)
8.1 Arithmetic Functions
269(10)
8.2 Higher Degree Congruences
279(10)
8.3 Quadratic Residues and Quadratic Reciprocity
289(22)
9 Structure Theory of Groups
311(42)
9.1 Group Actions, Permutation Representations
311(10)
9.2 Sylow Theorems
321(14)
9.3 Finite Abelian Groups
335(3)
9.4 Normal Series and Composition Series
338(15)
10 Structure Theory Continued
353(34)
10.1 Decompositions of Groups
353(5)
10.2 Solvable Groups
358(7)
10.3 Nilpotent Groups
365(12)
10.4 Free Groups and Presentations of Groups
377(10)
11 Arithmetic in Rings
387(34)
11.1 Division in Rings
387(6)
11.2 Principal Ideal Domains
393(6)
11.3 Euclidean Domains
399(5)
11.4 Chinese Remainder Theorem in Rings
404(2)
11.5 Unique Factorization Domain (U.F.D.)
406(15)
Appendix 421(8)
Index 429
RAMJI LAL is an adjunct professor at the Harish-Chandra Research Institute (HRI), Allahabad, Uttar Pradesh. He started his research career at the Tata Institute of Fundamental Research (TIFR), Mumbai, and served the University of Allahabad in different capacities for over 43 years: as a professor, head of the department and the coordinator of the DSA program. He was associated with HRI, where he initiated a postgraduate (PG) program in mathematics and coordinated the Nurture Program of National Board for Higher Mathematics (NBHM) from 1996 to 2000. After his retirement from the University of Allahabad, he was an advisor cum adjunct professor at the Indian Institute of Information Technology (IIIT), Allahabad for over three years. His areas of interest include group theory, algebraic K-theory and representation theory
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