Math 417 - Introduction to Abstract Algebra - Fall 2019
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: Mondays and Fridays 1.30-2.30 pm (or by appointment);
Class meets: MWF 11:00-11:50 am, 341 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:
Chapters 1-6 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.
- Frederick M. Goodman, Algebra: Abstract and Concrete (Edition 2.6), SemiSimple Press Iowa City, IA. [PDF file available for download free of charge]
Other Textbooks: (first ref available on the web; last two refs on hold in the Math Library)
- Manuel Ricou and Rui L. Fernandes, Introduction to Algebra [these are my notes that cover the same material but in different order; may contain typos]
- Michael Artin, Algebra, (2nd edition) Prentice Hall, 1991. [some level as the recommended textbook with alternative approaches]
- Garrett Birkhoff and Saunders MacLane, A survey of modern algebra , (4th Edition), Macmillan, 1977. [a classic book; more advanced than the recommended textbook]
There will be weekly homework/quizzes, 3 midterms and a final exam. All exams/midterms will be closed book.
- 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 Monday class. No late homework will be accepted. There will be 13 homework assignments, but 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 in class on Friday Sep 20, Friday Oct 18 and Friday Nov 15 (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 Thursday, Dec 19, 1.30 pm, in the regular classroom.
- Homework #1: Goodman, Exercises 1.3.1, 1.3.2, 1.3.3, 1.4.1, 1.4.2, 1.4.3, 1.5.1, 1.5.2, 1.5.3. [Important Note: In the textbook, the only symmetries considered are rigid motions, i.e., rotations+translations].
- Homework #2: Goodman, Exercises 1.5.5, 1.5.6, 1.5.8, 1.5.9, 1.5.10, 1.7.4, 1.7.5, 1.7.6, 1.7.8, 1.7.9.
- Homework #3: Goodman, Exercises 1.6.3, 1.6.4, 1.6.7, 1.6.8, 1.6.9, 1.10.1, 1.10.2, 1.10.3, 1.10.4.
- Homework #4: Goodman, Exercises 1.10.5, 1.10.6, 2.1.2, 2.1.4, 2.1.5, 2.1.6, 2.1.7.
- Mock midterm 1 to practice.
- Homework #5: Goodman, Exercises 2.1.10, 2.1.11, 2.1.12, 2.2.2, 2.2.4, 2.2.6, 2.2.9, 2.2.11.
- Homework #6: Goodman, Exercises 2.2.14, 2.2.15, 2.2.16, 2.2.17, 2.2.19, 2.3.5, 2.3.7, 2.3.8, 2.4.5, 2.4.8, 2.4.9.
- Homework #7: Goodman, Exercises 2.4.13, 2.4.17, 2.4.18, 2.5.4, 2.5.6, 2.5.7, 2.5.8, 2.5.13, 2.5.14, 2.6.1, 2.6.3, 2.6.5.
- Homework #8: Goodman, Exercises 2.6.6, 2.7.3, 2.7.4, 2.7.6, 2.7.7, 3.1.9, 3.1.10, 3.1.13.
- Mock midterm 2 to practice.
- Homework #9: Goodman, Exercises 5.1.3, 5.1.5, 5.1.7, 5.1.8, 5.1.9, 5.1.12, 5.1.13, 5.1.14, 5.1.15.
- Homework #10: Goodman, Exercises 5.2.2, 5.2.4, 6.1.1, 6.1.4, 6.1.6, 6.1.10, 6.1.11, 6.2.1, 6.2.3, 6.2.4.
- Homework #11: Goodman, Exercises 6.2.7, 6.2.8, 6.2.9, 6.2.10, 6.2.11, 6.2.16, 6.2.19, 6.3.2, 6.3.7, 6.3.8.
- Homework #12: Goodman, Exercises 6.3.1, 6.3.3, 6.3.5, 6.3.6, 6.3.9, 6.3.10, 6.3.11.
- Mock midterm 3 to practice.
Sections of the book covered so far: 1.1-1.7, 1.10 (midterm 1 stops here), 2.1-2.7 (midterm 2 stops here), 3.1, 5.1, 5.2, 6.1-6.3 (midterm 3 stops here at the Isomorphism Theorem for Rings).
(PDF files can be viewed using Adobe Acrobat Reader which can be downloaded for free from Adobe Systems for all operating systems.)
Frequently Asked Questions about Homework
- How many homework assignments will there be?
At least 12, but eventually there will be more, depending on the weeks
when there are midterms.
- How is the grades of the homework calculated?
Only the ten highest grades will count for your total homework grade.
The homework grade will be the average of the ten best grades.
- How is each homework assignment graded?
Each homework assignment receives a grade in the range 0-10 (0 minimum,
10 maximum). The grader will only grade 3 or 4 problems from each
homework set (the same problems for all students).
- Can I turn in late homework?
- Can I turn in homework via e-mail?
- What if I fall sick, or what if I decide to go to Las Vegas to get
married, or... , and I cannot turn in homework on time?
One of the reasons that only the 10 best grades will count is to
accommodate for such situations.
For important emergency information related to fires, tornados or active threats, please look at the following leaflet.
Last updated November 11, 2019