Lecture Notes:   Groups   Rings and Fields (updated 8/12)
Abstract algebra is about groups, rings, and fields, and is basically impossible. But if you can make any sense of it, you might find it slightly interesting. Groups are hopelessly complicated, unless they're finite and abelian, in which case they're insultingly simple. Rings seem very dry, but eventually turn out to be connected to geometry, which makes them interesting; but it takes years to get there, so why bother? Fields seemed pretty manageable, until Galois related them to groups again, which made them hopelessly complicated. In any case, abstract algebra has had a profound effect on our culture, as the following figures show:
PrerequisitesMath 54, officially, but math 55 may also be helpful.
TextbookThere is no required textbook for the class - You can use the notes George and I wrote up as the textbook. However, you are encouraged to consult other sources. Some texts I like are "Algebra" by Artin, "Topics in Algebra" by Herstein, and "Abstract Algebra" by Dummit and Foote.
GradingThe grade is out of 400 points, distributed as follows: 125 points for each of two exams, 120 points for HW (8 assignments at 15 points each), and 30 points for class participation (coming to lectures, office hours, etc).
You must take at least one exam to pass the course. In the event of a serious medical emergency, you may miss ONE exam if you have written documentation of your illness. You cannot miss both exams.
HomeworkHomework is due every Monday (and also Wednesday on an exam week), either in class or under my office door by 4PM. No late Hw will be accepted, so if you can't finish just submit what you have so far.
In doing your homework, you should work with others, but please write the names of your collaborators at the top of the assignment. You must write clearly - points will be taken off if your explanations are confusing or illegible. When writing proofs, you must use complete sentences.
ExamsThere are two exams, on July 17th and August 14th, weighted equally at 125 points. Both will take the entire class period and will be held in 70 Evans as usual.
LecturesWe meet for two hours each session. The first fifty minutes will be a lecture. After a ten minute break, the second hour will be a problem session.
Algebra is the study of operations, rules and procedures to solve equations. The origin of the term 'Algebra' seems to go back to a IX Century treaty by an Arab mathematician with the title 'The Compendious Book on Calculation by al-jabr and al-muqabala'. The term al-jabr is used in this book to denote two procedures: (i) the sum of two positive quantities to both sides of an equation, in order to cancel negative terms and (ii) the multiplication of both sides of an equation by a positive number to cancel fractions. With the passage of time, the term al-jabr or algebra became synonymous of the general study of equations and operations on them.
Algebra is one of the pillars of Mathematics and this course gives an introduction to the basics of Algebra, including conceptual proofs of all main results.
Lecturer:Rui Loja Fernandes
Email: ruiloja (at) illinois.edu
Office: 346 Illini Hall
Office Hours: M 10:00-10:50 am and Th 1.30-2.30 pm (or by appointment);
Class meets: MWF 09:00-09:50 am, 447 Altgeld Hall;
Prerequisites: Officially either MATH 416 or one of MATH 410, MATH 415 together with one of MATH 347, MATH 348, CS 373; or consent of instructor. In practice, ability to understand and write proofs.
In this page:
- I have posted the final grades. You can check it in the Math Department on-line grading system. To figure out your final grade, you have to do the following:
- Choose your top 10 HW grades, add them (gives a number between 0-100) and multiply by 1.5 (giving a number between 0-150);
- Add the grade of the homework (0-150), the 3 midterms (0-300) and the final (0-300). This gives a score between 0-750. Here is the graph of the total grades.
- The curve is a follows: A+>675, 675>A>630, 630>A->600, 600>B>550, 550>B->520, 520>C>460, 460>C->380, 380>D>300, 300>F.
- Here is a solution of the Final Exam. Grades will be posted on-line by 7 pm, Sunday, May 6. To see you final or for any other issues with grades, you can come by my office, Monday, May 7, 10-11.50 am.
- I have made available Homework #13. It is due Friday, May 5, at the beginning of the Final Exam.
- I have posted the grades of the 3rd midterm. They are available in Math Department on-line grading system. Here is a graph of the grades of the 3 midterms and 10 best homework (so far) and here is a solution of the 3rd midterm. The curve is as follows: A>90, B>75, C>60, D>50.
- I have posted the grades of the 2nd midterm. They are available in Math Department on-line grading system. Here is a graph of the grades of the two midterms and homework and here is a solution of the 2nd midterm. The curve is as follows: A>89, B>74, C>59, D>49.
- I have posted the grades of the 1st midterm. They are available in Math Department on-line grading system. Here is a graph of the grades and here is a solution of the 1st midterm. The curve is as follow: A>89, B>74, C>61, D>49.
- I have changed one of my office hours due to schedule conflicts with many students.
- Students can view the scores of homework on-line (please note that you will have to log in).
- Important information concerning the drop deadlines for this course can be found here.
- Chapters 1-3 of the recommended text, covering: Fundamental theorem of arithmetic, congruences. Permutations. Groups and subgroups, homomorphisms. Group actions with applications. Polynomials. Rings, subrings, and ideals. Integral domains and fields. Roots of polynomials. Maximal ideals, construction of fields.
- Manuel Ricou and Rui L. Fernandes, Introduction to Algebra (these notes will be updated frequently to correct typos and include new parts)
- Frederick M. Goodman, Algebra: Abstract and Concrete (Edition 2.6), SemiSimple Press Iowa City, IA. 
- Michael Artin, Algebra, (2nd edition) Prentice Hall, 1991. 
- Garrett Birkhoff and Saunders MacLane, A survey of modern algebra , (4th Edition), Macmillan, 1977. 
- Homework and quizzes (20% of the grade): Homework problems are to be assigned once a week. They are due the following week, at the beginning of the Friday class. No late homework will be accepted. Only the ten best grades will count, and the other homework grades will be dropped. If necessary, quizzes may be offered during the semester.
- Midterms (40% of the grade): The midterms will take place on February 17, March 17 and April 21 (the dates are subject to change).
- Final Exam (40% of the grade): You have to pass the final to pass the course. According to the non-combined final examination schedule it will take place Friday, May 5, 7-10 pm, in the regular classroom.
- Homework #1: Read Sections 1.1 and 1.2 of the lecture notes and solve Exercises 1.1.1, 1.1.2, 1.1.3, 1.1.6, 1.1.7, 1.2.1, 1.2.2, 1.2.4.
- Homework #2: Read Sections 1.3 and 1.4 of the lecture notes and solve Exercises 1.2.8, 1.2.11, 1.2.13, 1.3.1, 1.3.2, 1.3.4, 1.3.9, 1.4.3, 1.4.4.
- Homework #3: Read Section 1.4 of the lecture notes and solve Exercises 1.3.7, 1.4.5, 1.4.7, 1.4.8, 1.4.10, 1.4.11, 1.4.13.
- Homework #4: Practice for the 1st Midterm by solving the Mock Midterm 1, read Section 1.5 and solve Exercises 1.5.1, 1.5.2, 1.5.6, 1.5.7, 1.5.10, 1.5.12, 1.5.14.
- Homework #5: Read Section 1.6 of the lecture notes and solve Exercises 1.6.3, 1.6.4, 1.6.5, 1.6.9, 1.6.12, 1.6.15, 1.6.16.
- Homework #6: Read Sections 1.7 and 1.8 of the lecture notes and solve Exercises 1.7.1, 1.7.3, 1.7.6, 1.7.8, 1.7.9, 1.8.1, 1.8.5, 1.8.6, 1.8.7.
- Homework #7: Read Sections 2.1 and 2.2 of the lecture notes and solve Exercises 2.1.2, 2.1.5, 2.1.7, 2.2.2, 2.2.4, 2.2.6, 2.2.7, 2.2.8.
- Homework #8: Practice for the 2nd Midterm by solving the Mock Midterm 2, read Sections 2.3 and 2.4 of the lecture notes and solve Exercises 2.3.1, 2.3.4, 2.3.5, 2.4.1, 2.4.3, 2.4.5, 2.4.6, 2.4.8, 2.4.9.
- Homework #9: Read Sections 2.5 and 2.6 of the lecture notes and solve Exercises 2.5.1, 2.5.4, 2.5.5, 2.5.6,2.5.7, 2.5.9.
- Homework #10 Read Sections 2.6 and 2.7 of the lecture notes and solve Exercises 2.6.2, 2.6.3, 2.6.4, 2.6.5, 2.6.7, 2.6.10, 2.6.11, 2.6.15.
- Homework #11 Read Sections 2.8 and 2.9 of the lecture notes and solve Exercises 2.7.1, 2.7.6, 2.7.11, 2.8.1, 2.8.3, 2.8.6, 2.8.8, 2.8.9, 2.8.10, 2.8.14, 2.8.16.
- Homework #12 Read Sections 3.1 of the lecture notes and solve Exercises 3.1.3, 3.1.5, 3.1.9, 3.1.11, 3.1.12, 3.1.15, 3.1.19.
- Homework #13 Read Sections 4.1 and 4.2 of the lecture notes and solve Exercises 4.1.2, 4.1.4, 4.1.5, 4.1.6,4.1.11, 4.1.13, 4.2.1, 4.2.2, 4.2.3, 4.2.4..
Grading Policy and ExamsThere will be weekly homework/quizzes, 3 midterms and a final exam. All exams/midterms will be closed book.
Homework Assignments and Sections covered so far:
Sections of the lecture notes covered: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.1, 4.1, 4.2.
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