MATH 3140

SYLLABUS AND HOMEWORK

MATH 3140
Fall 2016

Abstract Algebra 1

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Homework is due in class and must be stapled, with your name and homework number on it, to receive credit.

You may find the Mathematics Help Room (MATH 175) to be useful as a meeting point for discussing homework.

Date	Topics covered	Reading	Homework Assignments
Monday August 22	Review of prequisites Sets, relations, maps, equivalence relations, and POSETS.	Fraleigh Section 0 The Book of Proof, Richard Hammack, Creative Commons, 2nd edition, 2013. (Available for free at the author's website.) Judson Chapter 1 and Chapter 2
Wednesday August 24	Review of prerequisites continued Linear agebra.	Linear Algebra and its applications, by David C. Lay, 4th edition, Chapters 1--5. Calculus, Volume II, by Tom M. Apostol, 2nd edition. The following .pdf gives a brief overview of vectors spaces and linear maps. Fraleigh Appendix: Matrix Algebra
Friday August 26	Introduction to groups of matrices Examples of matrix groups and permutation groups.		Homework 1: Due Friday August 26 Fraleigh Exercises 0: 2, 4, 10, 12, 14, 18, 28, 30, 36. Exercises on the following .pdf: 6.1.4, 6.1.7, 6.3.18, 6.3.25. Fraleigh Exercises A (Appendix): 2-14 even. Here is a LaTeX template for your homework, if you want .pdf, .tex, .bib.
Monday August 28	Introduction to groups: part II Abstract defintion of a group, subgroups, products of groups.	Fraleigh Part I, Fraleigh Sections 1-5 Judson Chapter 3
Wednesday August 31	Homomorphism of groups Definition of a homomorphism, image, kernel and isomorphism.	Fraleigh Part III, Fraleigh Section 13 Judson Chapter 11
Friday September 2	PREREQUISITE EXAM	PREREQUISITE EXAM	Homework 2: Due Friday September 2 Fraleigh Exercises 1: 4, 9, 12, 17, 37. Fraleigh Exercises 2: 2-24 even, 26-37. Fraleigh Exercises 3: 4, 8, 12, 16, 24-28. Here is a sample solution to an exercise (not to be graded .pdf, .tex). Prepare for PREREQUISITE EXAM
Monday September 5	NO CLASS	LABOR DAY	NO CLASS
Wednesday September 7	Matrix representations of groups Definitions and basic properties	Artin Chapter 9, Sections 1 and 2 Judson Chapter 12
Friday September 9	Cyclic groups Elementary properties, structure, subgroups.	Fraleigh Section 6 Judson Chapter 4	Homework 3: Due Friday September 9 Fraleigh Exercises 13: 1-5 odd, 12, 13. Fraleigh Exercises 4: 2, 4, 11, 19, 29, 30, 31. Fraleigh Exercises 5: 1-3, 8, 13, 14, 43, 45, 46, 51, 54. Not for credit: Fraleigh Exercises 13: 21, 47, 48.
Monday September 12	Generating sets Definition of a generating set, intersections of subgroups.	Fraleigh Section 7
Wednesday September 14	Groups of permutations Permutation groups, symmetric groups, Cayley's theorem.	Fraleigh Part II, Fraleigh Section 8 Judson Chapter 5 You can also read the following note I wrote in class on Cayley's Theorem, as a demonstration of LaTeX .pdf, .tex.
Friday September 16	Orbits, cycles, and the alternating group Orbits, cycles, length, parity, alternating groups.	Fraleigh Section 9	Homework 4: Due Friday September 16 Fraleigh Exercises 6: 5, 18, 30, 32, 50. Fraleigh Exercises 7: 2, 6.
Monday September 19	Cosets and the theorem of Lagrange Left and right cosets, the Theorem of Lagrange.	Fraleigh Section 10 Judson Chapter 6
Wednesday September 21	Finitely generated abelian groups Products of cyclic groups, order of elements, the Fundamental Theorem of Finitely Generated Abelian Groups.	Fraleigh Section 11 Judson Chapter 13
Friday September 23	Factor groups Normal subgroups, definition of the factor group, quotient groups and homomorphisms.	Fraleigh Section 14 Judson Chapter 10	Homework 5: Due Friday September 23 Artin Chapter 9 Exercises 1.2, 2.2. Fraleigh Exercises 8: 1-5 odd, 9, 21, 30, 34, 35. Fraleigh Exercises 9: 2, 10, 18, 23, 27. Fraleigh Exercises 10: 3, 5, 19, 35, 39, 40. Not for credit: Fraleigh Exercises 8: 36, 44, 49. Fraleigh Exercises 9: 29, 31, 33, 34, 36 (sample solution to 9.33 .pdf, .tex). Fraleigh Exercises 10:30, 31, 32, 33, 45.
Monday September 26	Homework review and introduction to LaTeX	Read through the materials at the bottom of this webpage on using LaTeX. If you have a laptop, try to install LaTeX on your laptop before class, and bring your laptop to class. If you do not have a laptop, that is not a problem.	Let me know in advance if there are particular homework problems you would like to review.
Wednesday September 28	Review	Review the practice exam.
Friday September 30	MIDTERM I	MIDTERM I	MIDTERM I Practice exam (solutions)
Monday October 3	Factor groups continued Simple groups, center, and commutator subgroup.	Fraleigh Section 15
Wednesday October 5	Rings and fields Definitions, basic properties, and rings of matrices.	Fraleigh Part IV, Fraleigh Section 18 Judson Chapter 16.1
Friday October 7	Rings and fields continued Homomorphisms, isomorphisms, and fields.		Homework 6: Due Friday October 7 Fraleigh Exercises 11: 10, 16, 17, 24, 36, 46, 47, 54 (sample solution to 11.47 and 13.47 .pdf, .tex). Fraleigh Exercises 14: 5, 6, 21, 26, 27, 30, 37, 38, 39 (sample solution to 14.30 .pdf, .tex) 2 points extra credit for typing in LaTex
Monday October 10	Integral domains Zero divisors, cancellation, integral domains, characteristic.	Fraleigh Section 19 Judson Chapter 16.2
Wednesday October 12	Fermat's and Euler's Theorems Fermat's Theorem, Euler's Theorem, applications.	Fraleigh Section 20 Judson Chapter 6.3
Friday October 14	Fraction fields The ring of fractions of an integral domain, and the total ring of fractions of a commutative ring	Fraleigh Section 21 Judson Chapter 18.1	Homework 7: Due Friday October 14 Fraleigh Exercises 15: 1, 4, 8, 10, 15, 34, 35, 36, 37, 39. Fraleigh Exercises 18: 2, 9, 23, 32, 33, 42, 46 (sample solution to 18.42 .pdf, .tex). Fraleigh Exercises 19: 1, 2, 6, 10, 15, 24, 29. 2 points extra credit for typing in LaTex
Monday October 17	Polynomial rings Polynomial rings in one variable, and evaluation homomorphisms.	Fraleigh Section 22 Judson Chapter 17.1
Wednesday October 19	Factorization of polynomials over a field The division algorithm over a field, irreducible polynomials, uniqueness of factorization.	Fraleigh Section 23 Judson Chapter 17.2
Friday October 21	Homomorphisms and factor rings Homomorphisms, properties of homomorphisms.	Fraleigh Part V, Fraleigh Section 26 Judson Chapter 16.3	Homework 8: Due Friday October 21 Fraleigh Exercises 20: 2, 5, 9, 23, 29. Fraleigh Exercises 21: 1, 4, 12, 13. Fraleigh Exercises 22: 3, 5, 7, 14, 23, 24, 27, 28. 2 points extra credit for typing in LaTex
Monday October 24	Homomorphisms and factor rings continued Factor and quotient rings.
Wednesday October 26	Review
Friday October 28	Homework review		Homework 9: Due Friday October 28 Fraleigh Exercises 23: 2, 7, 14, 16, 25, 34, 35. Fraleigh Exercises 26: 1, 4, 17. 2 points extra credit for typing in LaTex
Monday October 31	Review	Review the practice exam.
Wednesday November 2	Homework review
Friday November 4	MIDTERM II	MIDTERM II	MIDTERM II Practice exam (solutions)
Monday November 7	Prime and maximal ideals Prime and maximal ideals, prime fields.	Fraleigh Section 27 Judson Chapter 16.4
Wednesday November 9	Prime and maximal ideals continued Ideals in polynomials rings in one variable over a field, and unique factorization.
Friday November 11	Grobner bases Algebraic varieties and polynomial rings, illustrations.	Fraleigh Section 28 A more thorough treatment can be found in D. Cox, J. Little, and D. O'Shea, Ideals, varieties, and algorithms, Fourth edition. Undergraduate Texts in Mathematics. Springer, 2015.	Homework 10: Due Friday November 11 Fraleigh Exercises 26: 20, 21, 28, 29, 30, 34. Fraleigh Exercises 27: 2, 5, 24, 27, 31, 32, 33, 34, 35. 2 points extra credit for typing in LaTex
Monday November 14	Grobner bases continued Grobner bases.
Wednesday November 16	Extension fields Introduction to extension fields, algebraic and transcendental elements, irreducible polynomials of elements, and simple extensions.	Fraleigh Part VI, Fraleigh Section 29 Judson Chapter 22.1
Friday November 18	Review		Homework 11: Due Monday November 28 Fraleigh Exercises 28: 2, 6, 15, 16, 19. Not for credit: Fraleigh Exercises 28: 20, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36. 2 points extra credit for typing in LaTex
Monday November 21 - Friday November 25	THANKSGIVING BREAK	THANKSGIVING BREAK	THANKSGIVING BREAK
Monday November 28	Algebraic field extensions Finite extensions and algebraic extensions.	Fraleigh Part VI, Fraleigh Section 31
Wednesday November 30	Review
Friday December 2	Homework review		Homework 12: Due Friday December 2 Fraleigh Exercises 29: 3, 4, 7, 13, 23. 2 points extra credit for typing in LaTex
Monday December 4	Review
Wednesday December 6	Homework review
Friday December 9	Review		Homework 13: Review for the final exam
Thursday December 15	FINAL EXAM 7:30 PM -- 10:00 PM HLMS 181 (Lecture Room)	FINAL EXAM	FINAL EXAM Practice exam (solutions)

*An asterix indicates that the assignment has not been finalized (and may be changed)

I strongly encourage everyone to use LaTex for typing their homework. If you have a mac, one possible easy way to get started is with texshop. If you are using linux, there are a number of other possible ways to go, using emacs, ghostview, etc. If you are using windows, you're on your own, but I'm sure there's something online. Here is a sample homework file to use: (the .tex file, the .bib file, and the .pdf file). It is probably easiest to just look at the .tex file, and start to experiment. There is also the (Not so) short introduction to latex, which will answer most of your questions (although typing your question into your favorite search engine will probably work well, too). This site can help you find LaTeX symbols by drawing: http://detexify.kirelabs.org/classify.html. You may also want to try https://cloud.sagemath.com/ or https://www.sharelatex.com/ for a cloud version.