Text:  Thomas Hungerford "Algebra", ISBN 0-387-90518-9 or 3-540-90518-9
 

Office Hours: By appointment. Other good times to talk are right before or right after the class. It is also possible to ask questions by email, which will be generally answered within 24 hours, often sooner. The more detailed your email question is, the more detailed the reply message will be. My email is borisov@rutgers.edu.
 

Final/midterm Examinations: There will not be any.
 

Homework: There will be an extensive homework requirement. An average of five homework problems will be assigned after each class. I will collect a week worth of homework each Thursday, starting from January 29. Collaboration on homework assignemnts is allowed and strongly encouraged, but each person must write up their solutions. Homework can be handwritten or typed up. If you can not make it to class when homework is due, you can take photos of your homework and email me at borisov@math.rutgers.edu. You can also submit homework by email if that is your preference.
 

Weather/Covid rules: If in-person classes are canceled, we will move online to our Canvas site.
 

Schedule of Lectures (updated periodically). Please try to read the relevant textbook sections before the lecture.

January 22 Section 5.1
Homework: page 241; 6 (hint: use the fact that the polynomial ring K[x_1,...,x_n] is a UFD), 8, 14, 17, 18.

January 26 Section 5.2
Homework: page 255; 2, 5, 7, 11, 15.

January 29 homework from the last two lectures is due
Section 5.3
Homework: page 254; 3(a), 5, 8, 20, 23.

February 2 Sections 5.4, 5.5
Homework: page 277; 6, 12; page 281; 3, 6, 8.

February 5 homework from the last two lectures is due
Sections 5.7, 5.8
Homework: page 296; 4, 8; page 300; 8, 9;
Problem: Calculate the n=10 cyclotomic polynomial over Q.

February 9 Sections 5.9
Homework: page 309; 1, 3, 4;
Problem: Suppose that an irreducible cubic polynomial over Q has three real roots. Prove that it is impossible to have a formula for these roots involving repeated radicals, starting with rational numbers so that all intermediate radicals are real numbers.
Problem: Suppose that a complex number x is an algebraic integer (over Z) and is such that all of its Galois conjugates have length 1. Prove that x is a root of unity.

February 12 homework from the last two lectures is due
Commutative rings -- examples. Prime and maximal ideals. Jacobson and Nil radicals. Localization.
Homework:
Problem 1. Prove (directly, without talking about radicals) that if x is a nilpotent element of a ring R, then 1+xy is invertible for all y.
Problem 2. Give an example of a ring whose Jacobson and Nil radicals are different.
Problem 3. Give an example of a ring homomorphism R -> T and a maximal ideal of T whose preimage in R is not maximal.
Problem 4.

February 16 Noetherian rings and modules. Local rings. Nakayama's Lemma.
Homework

February 19 homework from the last two lectures is due
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February 23 Sections
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February 26 homework from the last two lectures is due
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March 2 Sections
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March 5 homework from the last two lectures is due
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March 9

March 12 homework from the last two lectures is due


March 23 Sections
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March 26 homework from the last two lectures is due
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March 30 Sections
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April 2 homework from the last two lectures is due
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April 6 Sections
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April 9 homework from the last two lectures is due
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April 13 Sections
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April 16 homework from the last two lectures is due
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April 20 Sections
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April 23 homework from the last two lectures is due
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April 27 Sections
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April 30 homework from the last two lectures is due
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May 4 Sections
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