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:**

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

**Announcements:**

- 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.

**Syllabus:**

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.

**Textbooks:**

**Recommended Textbook:**

- Manuel Ricou and Rui L. Fernandes,
(these notes will be updated frequently to correct typos and include new parts)**Introduction to Algebra**

**Other Textbooks:** (first ref available on the web; last two refs on hold in the Math Library)

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

**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..

**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.**

(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 April 29, 2017*