Part II - Rings

Lecture 8: Rings, Integral Domains, and Euler's Theorem

IV: 18–20

 Jan 11, 10–12
E32

Suggested exercises:

18: 6, 8, 11, 20, 22, 25, 38, 44
19: 7, 11, 25, 26
20: 1, 4, 8, 10

Problem set 8
due Jan 18, hand in with Ornella Greco

Lecture 9: Quotient fields and polynomial rings

IV: 21–23

 Jan 15, 13–15
E32

Suggested exercises:

21: 2, 12–17
22: 5, 13, 22, 25, 27
23: 1, 3, 9, 34, 36

 Problem set 9
due Jan 24, hand in with Ornella Greco or in class
Exercises

Jan 21, 15–17
E32

    

Lecture 10: Irreducibility
Ideals, quotient rings, prime and maximal ideals

IV: 23, V: 26–27

 Jan 24, 8–10
E32

Suggested exercises:

23: 7, 12, 14, 21
26: 1, 3, 17, 30
27: 2, 6, 24, 34–37

Problem set 10
due Jan 31, hand in with Ornella Greco
Exercises Jan 28, 15–17
E53

 

 

  

Lecture 11: PIDs and unique factorization domains

IX: 45

Feb 1, 15–17
E32

Suggested exercises:

26: 31–36
27: 33
45: 4, 8, 10, 28

 Problem set 11
due Feb 7, hand in with Ornella Greco
Exercises

Feb 7, 10–12
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Lecture 12: Euclidean domains; modules over rings

IX: 46–47, notes Section 1

 Feb 8, 15–17
E32

Suggested exercises:

3, 5, 6

 Problem set 12 = Ex. 1, 2, 4 in the notes. Due Feb 15, hand in with Afshin Goodarzi
Exercises Feb 12, 15–17
D41		    

Lecture 13: free modules and finitely generated modules; the structure of finitely generated modules over PIDs I

notes Section 1–2

Feb 14, 10–12
E32

Suggested exercises:

7, 9, 12

 Problem set 13 = Ex. 8, 10, 11.
Due Feb 21, hand in with Afshin Goodarzi
Exercises Feb 19, 10–12
E32		    

Lecture 14: the structure of finitely generated modules over PIDs II; the Jordan normal form

notes Section 2–3

Feb 22,13–15
E53

Suggested exercises:

13, 14, 17, 18, 20

 Problem set 14 = Ex. 15, 16, 19.
Due Mar 1, hand in with Afshin Goodarzi
Exercises Feb 28, 10–12
E32		    
Exercises Mar 7, 10–12
E32		    
Exam Mar 11, 14–17