Math 5320 Spring 2018
The course syllabus is available here.
A few additional organizational notes:
Collaboration with other students on homework is permitted and encouraged, but write your solutions individually and please make a note of who you worked with.
The definitions of various terms given in class won't typically conflict with those given in the textbook, but if they do, the ones in class should take precedence.
Reading, past worksheets, calendar for the course
Part 1 of the course will be concerned with defining and exploring the properties of rings, subrings, fields, ring homomorphisms, ideals and quotient rings, product rings, maximal ideals and prime ideals, among other things. The first portion of the course will be drawn from Ch. 11 of the textbook, though some portions of that chapter will not be covered. In particular the emphasis is on sections 11.1-11.4, 11.6 and 11.8.
Worksheet 1, Jan. 8
Worksheet 2, Jan. 10
Worksheet 3, Jan. 12
Worksheet 4, Jan. 17
Worksheet 5, Jan. 19
Worksheet 6, Jan. 22
Worksheet 7, Jan. 24
Worksheet 8, Jan. 29
Midterm 1 was on Jan. 31 and covered material from Part 1 up to our discussion of the Chinese Remainder Theorem.
Worksheet 9, Feb. 2
Part 2 of the course started roughly around Feb. 2. It is concerned with domains and factorization. Its topics include integral domains, Euclidean domains, principal ideal domains, unique factorization domains, and irreducibility. It is drawn mainly from Ch. 12, emphasizing sections 12.1-12.4.
Worksheet 10, Feb. 5
Worksheet 11, Feb. 7
Handout on Euclidean algorithm for the Gaussian integers, Feb. 7
Worksheet 12, Feb. 9
Worksheet 13, Feb. 12
Worksheet 14, Feb. 14
Worksheet 15, Feb. 16
Worksheet 16, Feb. 21
Worksheet 17, Feb. 23
Worksheet 18, Feb. 26
Midterm 2 was on Feb. 28 and covered material from products of rings up through our discussion of polynomial long division with monic divisors.
Worksheet 19, Mar. 2
Handout on factorization terminating, Mar. 5
Part 3 of the course starts Mar. 7 and will be concerned with field theory and cover parts of Ch. 15.
Worksheet 20, Mar. 7
Worksheet 21, Mar. 9
Worksheet 22, Mar. 12
Worksheet 23, Mar. 14
Worksheet 24, Mar. 16
Worksheet 25, Mar. 26
Handout on irreducibility of x^4+1, Mar. 29
Midterm 3 will be on March 30 and will cover material up through our discussion of the primitive element theorem on Monday Mar. 26.
Worksheet 26, Mar. 28
Worksheet 27, Apr. 2
Worksheet 28, Apr. 4
Worksheet 29, Apr. 6
Worksheet 30, Apr. 9
Worksheet 31, Apr. 11
Worksheet 32, Apr. 13
Worksheet 33, Apr. 16
Worksheet 34, Apr. 18
Worksheet 35, Apr. 20
Worksheet 36 , Apr. 23
The final exam will be on April 30, 1-3 pm. Here is a
Due Jan. 17: Ch.11: 1.6,1.7,1.8 (Hint: Use Bezout's identity), 1.9, 3.2, 3.8, 3.9. See the note at the end of 11.3 for a dicussion of characteristic.
Due Jan. 24: Ch.11: 2.2, 3.7, 3.10, 3.13, 4.3(b,c,d) (All of the final answers should be of the form Z/(nZ)), 6.8 (In part (d), not only produce the idempotents, but also show the isomorphisms between the ideals generated by each idempotent and the quotient they correspond to)
Due Feb. 7: Ch. 11: 6.1, 6.2, 7.1, 7.2, 8.2(a),(b) Hint for 7.1: Given any a in R, show that the *function* from the nonzero elements of R to themselves given by multiplication by a is injective.
Due Feb. 14: Ch. 11: 3.3(b), Ch. 12: 2.6(b) (Z[sqrt(-2)] is defined the same as the Gaussian integers but with sqrt(-2) in place of i. The discussion at the beginning of chapter 11.5 also shows how to define them.), 2.8 (Just the second part: you don't have to describe the systematic way.) Also, there are several problems not in the book for you to do:
Due Feb. 21: Ch. 11: 6.7, 8.2(c)-(d), 8.4, M.1, Ch. 12: 2.4
Due Feb. 26: Ch. 11: M.5, Ch. 12: 2.2, 2.10, 3.2 Some notes: The kernel in M.5 will be principally generated. It will probably be necessary to use our discussion of polynomial long division that will happen in class on 2/23. If you want to work ahead, take a look at Example 11.3.16 in the textbook. For Ch. 12 2.10, recall what the ideals of a formal power series ring are...
Due March 7: On this page Due March 14: Ch. 15: 1.1, 1.2, 2.1, 2.2, 3.1
Due March 28: The complete problem set is on this page It consists of a few book problems as well as a couple of other problems that are meant to help you with solving some of them. However, if you'd like to take a look at the problems in the book before seeing the "hints", they are: Ch. 15: 2.3, 3.2, 3.3, 3.4(a-c), 4.2(a-d)
Due April 4: Ch 15: 5.2(b), 5.3, 6.2 (possible hint: see Prop'n 15.3.3), 6.3, Ch. 16: 1(a-b) A fact that you can use to help prove 6.3: If z is a primitive nth root of unity and m is relatively prime with n, then z^m is also a root of the irreducible polynomial of z over Q. Due April 11: On this page
Due April 18: Ch. 16: 7.1,7.2,7.4,7.8,7.11 (For problem 7.1, it's fine to use the main theorem. Also, it's fine to assume that the square roots of a and b are not in F and that the square root of b is not contained in F adjoin the square root of a)
Optional, due April 30: Review Problems
- Use properties of Euclidean domains from lecture Monday Feb 5 to do the following problem. Let p(x) be a polynomial in F[x] where F is a field. Show that if a_1,...,a_n are distinct roots of p(x), then (x-a_1)(x-a_2)...(x-a_n) divides p(x). Hint: A proof of this was sketched during lecture on Wed. Feb 7.
- Find all the maximal ideals of C[x]. You may (and will need to) use the fact that any polynomial p(x) in C[x] has a root in C. Do not use the notion of irreducibility (which is what Artin does to prove it in 11.8). Instead prove this using the consequences of C[x] being a Euclidean domain from lecture on Monday Feb 5.
- Using the definition of characteristic from class on Friday Feb 9, prove that for any ring R of positive characteristic p, there is an injective ring homomorphism Z/p --> R.