Tentative Syllabus We will primarily study rings and fields in this quarter, starting with chapter 12. A rough outline will be given below.

Homework assignments

Homeworks need to be turned in on or before the stated date, usually a Friday. No homework needs to be turned in in weeks when a midterm is scheduled. However, some of the homework may also be part of the material being asked for the midterm. It is very important that you do the homework problems as most of the exam problems will be variations of homework problems.

for Jan 8: Chapter 12: 2, 6, 18, 22, 23, 26, 40/42, 46/48 (recall that 40/42 means Problem 40 in the 5th edition (on reserve) and Problem 42 in the seventh edition; if there is only one number, it is the same for both editions).

for Jan 15: Chapter 13: 6, 13, 14, 16, 22/24, 24/26(you may use that the square root of d is irrational if d is not a perfect square), 30, 41/45, 54/58 (Hint: recall Fermat's little theorem from last quarter). Chapter 14: 4, 10, 12, 18/21

for Jan 22: Chapter 14: 6, 27/31 (hint: first show that A is indeed an ideal and then show that R/A is isomorphic to the field R of real numbers, see definition in Chapter 15), 29/33, 30/34, 32/36, 50/56, Chapter 15: 9/11, 12/14, 33/35, 34/36, 45/51 and: Consider the ideals A= <2> and B= <8> in Z, the ring of integers. Show that the additive group A/B is isomorphic to Z_4, but that the ring A/B is NOT isomorphic to the ring Z_4.

for Jan 29: Chapter 16: 12, 22(consider the polynomial f-g), 28, 31/33 (use Fermat's Little Theorem), 38/40 and: Let F be a field. Show that there exist a, b in F such that x^2+x+1 divides x^43 + ax +b. Chapter 17: 6, 8, 10(b), 12, 14

for Feb 5: (need not be turned in) Chapter 17: 6, 8 (if you want to redo them), see problem below, start reviewing for midterm (see below for more details); I will put on a few more review exercises a bit later. Here are a few to get started: Chapter 14: 34/38, Chapter 15: 15/17, 16/18

Let f(x) and g(x) be primitive polynomials in Z[x] satisfying f(x) = ag(x) for a rational number a. Show that a= 1 or a= -1. (Hint: write a = r/q for r and q relative prime integers. Then qf(x)=rg(x). Find a contradiction if there was a prime number p which divides q (or r)

Midterm I: The first midterm will take place on Friday, February 5 in class. You are allowed one hand-written cheat sheet, but no calculators or other notes. The material will go until including Chapter 16. In particular, make sure you have worked with the basic definitions. Examples:

Is 6Z an ideal of Z, a prime ideal of Z, a maximal ideal of Z?

selected homework solutions

Let f(x) and g(x) be primitive polynomials in Z[x] satisfying f(x) = ag(x) for a rational number a. Show that a= 1 or a= -1. (Hint: write a = r/q for r and q relative prime integers. Then qf(x)=rg(x). Find a contradiction if there was a prime number p which divides q (or r)

Determine if the following polynomials are irreducible over Q: 5x3 + 7x + 2, 21x3 - 6x2+ 7x -2 (Consider the rational-root test, see Chapter 17: 25 which we proved in class)

for Feb 19: Chapter 18: 17, 18, 22, 23, 30

for Feb 26: see below for review exercises.

Midterm II: The second midterm will take place on Friday, February 26 in class. As in the previos midterm, you are allowed one hand-written cheat sheet, but no calculators or other notes. The material will go until including Chapter 18, as well as the few items of chapters 19 and 20 covered in the exercises listed above. In particular, you should know the notion of minimal polynomial and degree of a field extension. Emphasis will be on the material after chapter 16. Below are some more review exercises:

Review exercises: Chapter 17: 10, 11, 14, 22, Chapter 19: 22, Chapter 20: 1, 17 and

What is the Q-dimension of Q(5^{1/3}), where 5^{1/3} is the cubic croot of 5, and of Q(\sqrt{3+\sqrt{7}}); please ask if the notation is not clear. (Hint: try to find the minimal polynomials of these numbers).

for March 5 : Chapter 23: 3, 4, 6, 7, 9, 10, 14, 15,

problems of first midterm

solutions for first midterm

problems of second midterm

solutions for second midterm

Change in computation of final grade: As announced in class, there will be a second way to calculate the grade which should allow students to improve who did not do well on one or two of the midterms. For the second option, one midterm is dropped, and the final will count 60% instead. We will calculate the final score for both options, and pick the higher one.

Info for final: The final will take place on Friday, March 19 8am-11am in our class room. Please bring a blue book or paper. The usual rule apply: you are allowed to use one cheat sheet, normal size, but no calculators, books or other notes.

The material will go over the sections from which homework was assigned. You need not worry about material in sections which we did not cover in class. E.g. in Chapter 31, we did NOT cover coset decoding, in Chapter 20 we did NOT cover splitting fields or anything after that in that chapter. There were no homework assignments from chapter 21, but you should know the formula [K:F]=[K:E][E:f] for the degrees of field extensions (see Theorem 21.5). We have posted solutions for the two midterms. Please go over these solutions as well as over homework problems and related problems in the book. Please email and/or come to office hours if you have any questions.

Office hours in finals week: W4-5, Th 1-3; Michele: Th 11am-noon.