Donu Arapura - Abstract Algebra done Concretely Donu...

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Unformatted text preview: Abstract Algebra done Concretely Donu Arapura February 19, 2004 Introduction: I wrote these notes for my math 453 class since I couldn’t find a book that covered basic abstract algebra with the level and emphasis that I wanted. Rather than spending a lot of time on axiomatics and serious theorem proving, I wanted to spend more time with examples, simple applications and with making scenic detours. I may have gotten a bit carried away with the detours, and certainly there is more here than can be covered in a semester at a reasonable pace. However, a lot of these side topics (which I marked with a star) can be skipped to save time. In order to try to encourage students to play around with examples, I tried to include some Maple code now and then. But its role is secondary and it can be ignored without losing too much. Also since I wanted to emphasize the mathematics rather than the algorithms, I didn’t go out of my way to implement these things efficiently. If you find typos or more serious errors, send me email.- Donu Arapura ([email protected]) 1 Contents 1 Natural Numbers 5 1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Principles of Counting 9 2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Integers and Abelian groups 12 3.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Divisibility 17 4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5 Congruences 19 5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6 Linear Diophantine equations 22 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 7 Subgroups of Abelian groups 25 7.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 8 Commutative Rings 28 8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 9 A little Boolean Algebra* 32 9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 10 Fields 35 10.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 11 Polynomials over a Field 37 11.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 12 Quotients of Abelian groups 41 12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2 13 Orders of Abelian groups 43 13.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 14 Linear Algebra over Z p * 46 14.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 15 Nonabelian groups 50 15.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 16 Groups of Permutations 54 16.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 17 Symmetries of Platonic Solids 58 17.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 18 Counting Problems involving Symmetry* 62 18.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....
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Donu Arapura - Abstract Algebra done Concretely Donu...

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