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Unformatted text preview: Algebra I – MATH235 Course Notes by Dr. Eyal Goren McGill University Fall 2007 Last updated: November 27, 2007. c All rights reserved to the author, Eyal Goren, Department of Mathematics and Statistics, McGill University. Contents 1. Introduction 4 Part 1. Some Language and Notation of Mathematics 6 2. Sets 6 2.1. 8 3. Proofs: idea and technique 9 3.1. Proving equality by two inequalities 9 3.2. Proof by contradiction and the contrapositive 10 3.3. Proof by Induction 11 3.4. Prove or disprove 13 3.5. The pigeonhole principle 13 4. Functions 14 4.1. Injective, surjective, bijective, inverse image 15 4.2. Composition of functions 16 4.3. The inverse function 16 5. Cardinality of a set 17 6. Number systems 20 6.1. The polar representation 22 6.2. The Fundamental Theorem of Algebra 24 1 2 7. Fields and rings  definitions and first examples 26 7.1. Some formal consequences of the axioms 28 Part 2. Arithmetic in Z 30 8. Division 30 9. GCD and the Euclidean algorithm 31 9.1. GCD 31 9.2. The Euclidean algorithm 32 10. Primes and unique factorization 33 10.1. Further applications of the Fundamental Theorem of Arithmetic 36 Part 3. Congruences and modular arithmetic 39 11. Relations 39 12. Congruence relations 40 12.1. Fermat’s little theorem 43 12.2. Solving equations in Z /n Z . 44 12.3. Public key cryptography; RSA method 45 Part 4. Polynomials and their arithmetic 47 13. The ring of polynomials 47 14. Division with residue 48 15. Arithmetic in F [ x ] 48 15.1. Some remarks about divisibility in a commutative ring T 49 15.2. GCD of polynomials 49 15.3. The Euclidean algorithm for polynomials 50 15.4. Irreducible polynomials and unique factorization 51 15.5. Roots 55 15.6. Eisenstein’s criterion 56 15.7. Roots of polynomials in Z /p Z 56 Part 5. Rings 58 16. Some basic definitions and examples 58 17. Ideals 61 18. Homomorphisms 63 18.1. Units 66 19. Quotient rings 67 19.1. The quotient ring F [ x ] / ( f ( x )) 70 19.2. Every polynomial has a root in a bigger field 71 19.3. Roots of polynomials over Z /p Z 72 20. The First Isomorphism Theorem 72 20.1. Isomorphism of rings 72 20.2. The First Isomorphism Theorem 73 20.3. The Chinese Remainder Theorem 74 21. Prime and maximal ideals 77 Part 6. Groups 78 3 22. First definitions and examples 78 22.1. Definitions and some formal consequences 78 22.2. Examples 78 22.3. Subgroups 80 23. The permutation and dihedral groups 80 23.1. Permutation groups 80 23.2. Cycles 82 23.3. The Dihedral group 84 24. The theorem of Lagrange 85 24.1. Cosets 85 24.2. Lagrange’s theorem 86 25. Homomorphisms and isomorphisms 87 25.1. homomorphisms of groups 87 25.2. Isomorphism 88 26. Group actions on sets 89 26.1. Basic definitions 89 26.2. Basic properties 89 26.3. Some examples 91 27. The CauchyFrobenius Formula 92 27.1. Some applications to Combinatorics 93 28. Cauchy’s theorem: a wonderful proof 96 29. The first isomorphism theorem for groups 97 29.1. Normal subgroups 97 29.2. Quotient groups 98 29.3. The first isomorphism theorem29....
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This note was uploaded on 04/18/2008 for the course MATH 235 taught by Professor Goren during the Fall '07 term at McGill.
 Fall '07
 Goren
 Math, Statistics, Algebra

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