Math 417 – Introduction to Abstract Algebra – Spring 2017

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.

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, Introduction to Algebra (these notes will be updated frequently to correct typos and include new parts)

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

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

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.)

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Last updated April 29, 2017