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 15. 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): 214 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 15 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: 224 even, 2637. Fraleigh Exercises 3: 4, 8, 12, 16, 2428. 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: 15 odd, 12, 13. Fraleigh Exercises 4: 2, 4, 11, 19, 29, 30, 31. Fraleigh Exercises 5: 13, 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: 15 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) 