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:** Wednesday 11:00 AM (or by appointment);**Class meets:** TTh 9:30-10:50 am, 2 Illini 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:**

- Announcements
- Syllabus
- Textbooks
- Grading Policy and Exams
- Homework Assignments
- Emergency information for students in Mathematics courses

**Announcements:**

- I have posted the grades of the final exam, as well as the final grades in this course, here. They are also available in Math Department on-line grading system. Students who wish to look at the grading of the exam should show up in my office on May 12, at 4 pm. You may wish to look at the solution of the final exam.
- I have made available the last homework set Homework 12. It is due Thursday, May 7, before 11 am, in my office.
- I will hold extra Office Hours on Wednesday, May 7, Thursday, May 8, and Friday, May 9, 10am-11am.
- The last class of the semester, on May 6, will be a problem session. We will be going over the mock final and I will answer any questions from the students.
- You can view the scores of homework on-line (please note that you will have to log in). You can also view a graph with the grades of the 1st and 2nd Midterms and overall grade taking into account Homework 1-8 (rescaled to 100). Your current grade is approximately the following, according to your overall score: A>85, 85>B>70, 70>C>60.
- LAS and Engineering students enrolled in Math 417 are permitted to drop this course without penalty and without petition until Friday, April 17, 2015. LAS students who wish to drop should contact the LAS office and ask to be directed to a COAR user. Engineering students should contact the Engineering College Office, 206 Engineering Hall.

**Syllabus:**

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.

**Textbooks:**

**Recommended Textbook:**

- Frederick M. Goodman,
(Edition 2.6), SemiSimple Press Iowa City, IA. [PDF file available for download free of charge]**Algebra: Abstract and Concrete**

**Other Textbooks:** (on hold in the Math Library)

- Michael Artin,
, (2nd edition) Prentice Hall, 1991. [some level as the recommended textbook with alternative approaches]*Algebra* - Garrett Birkhoff and Saunders MacLane,
, (4th Edition), Macmillan, 1977. [a classic book; more advanced than the recommended textbook]*A survey of modern algebra*

**Grading Policy and Exams**

There will be weekly homework/quizzes, 2 midterms and a final exam. All exams/midterms will be closed book.

**Homework and quizzes (25% of the grade)**: Homework problems are to be assigned once a week. They are due the following week, at the beginning of the Thursday 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 (each 20% of the grade):**The midterms will take place on February 26 and April 9 (the dates are subject to change).**Final Exam (35% 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 Monday, May 11, 7-10 pm, in the regular classroom.

**Homework Assignments and Sections covered so far:**

**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, 1.5.7, 1.5.9, 1.5.10, 1.6.3, 1.6.4, 1.6.8, 1.6.11, 1.6.14.**Homework #3:**Goodman, Exercises 1.7.4, 1.7.8, 1.7.9, 1.7.11, 1.7.12, 1.7.14, 1.7.16, 1.8.4, 1.8.6, 1.8.7.**Homework #4:**Goodman, Exercises 1.8.10, 1.8.12, 1.8.13, 1.8.17, 1.8.18, 1.10.4, 1.10.7, 1.10.9, 1.10.10.- Mock Midterm 1 and list of important notions/concepts. Midterm 1 with solutions.
**Homework #5:**Goodman, Exercises 2.1.3, 2.1.4, 2.1.6, 2.1.8, 2.1.9, 2.1.10, 2.1.15, 2.2.2, 2.2.3.**Homework #6:**Goodman, Exercises 2.2.4, 2.2.8, 2.2.11, 2.2.12, 2.2.14, 2.2.15, 2.2.16, 2.2.17, 2.2.19, 2.2.22.**Homework #7:**Goodman, Exercises 2.3.5, 2.3.6, 2.4.5, 2.4.8, 2.4.9, 2.4.13, 2.4.14, 2.4.17, 2.4.18, 2.5.4, 2.5.6, 2.5.7, 2.5.8.**Homework #8:**Goodman, Exercises 2.5.13, 2.5.14, 2.6.1, 2.6.3, 2.6.5, 2.7.3, 2.7.6, 2.7.7, 3.1.9, 3.1.10, 3.1.13.- Mock Midterm 2 and list of important notions/concepts. Midterm 2 with solutions.
**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 6.1.2, 6.1.4, 6.1.6, 6.1.10, 6.1.11, 6.2.1, 6.2.3, 6.2.4, 6.2.7.**Homework #11:**Goodman, Exercises 6.2.8, 6.2.9, 6.2.10, 6.2.11, 6.2.16, 6.2.22, 6.3.1, 6.3.2, 6.3.7.**Homework #12:**Goodman, Exercises 6.3.8, 6.3.9, 6.3.10, 6.4.1, 6.4.4, 6.4.7, 6.4.8, 6.4.13, 6.4.16.- Concepts to know for the final exam and Mock Final .
**Final Exam with solutions.**

Sections of the book covered: 1.1-1.8, 1.10-1.12, 2.1-2.7, 3.1, 5.1, 6.1-6.4.

(PDF files can be viewed using Adobe Acrobat Reader which can be downloaded for free from Adobe Systems for all operating systems.)

Emergency information for students in Mathematics courses For important emergency information related to fires, tornados or active threats, please look at the following leaflet.*Last updated May 12, 2015*