Math 422: Abstract Algebra II -- Spring 2010 -- Schedule & Homework

Week of April 19 - 23, 2010
We continue with Galois Theory this week. We'll study examples of computing Galois groups. After stating the Fundamental Theorem of Galois Theory we'll use Galois groups to understand the intermediate fields of a Galois extension. Next week we'll look briefly at some applications of Galois Theory.
The problems below form Homework #12 and are all due Wednesday, April 28.
Monday:
  • Study: Sections 14.1, 14.2

  • Do:
    problems
Wednesday:
  • Study:
  • Do:


Friday:
  • Study:

  • Do:


Week of April 12 - 16, 2010
This week we begin Galois Theory. We will define Galois extensions and Galois groups, and take a look at examples of these objects.
The problems below form Homework #11 and are all due Monday, April 19.
Monday:
  • Study: Section 13.5, 14.1

  • Do:
    problems
Wednesday:
  • Study: Section 14.1
  • Do: Section 13.2 #18 and Section 14.1 #8


Friday:
  • Study:

  • Do:


Week of April 5 - 10, 2010
We begin our work towards Galois Theory this week. After establishing (with some hand waving) the existence and uniqueness of splitting fields and algebraic closures, we will consider separable and inseparable field extensions.
The problems below form Homework #10 and are all due Monday, April 12.
Monday:
  • Study: Section 13.4

  • Do:
    problems
Wednesday:
  • Study: Section 13.4, 13.5
  • Do:


Friday:
  • Study: Section 13.5

  • Do: Consider the finite field with nine elements. We know that the set of nonzero elements of this field form a group under multiplication. For this problem find a generator of this group and show, explicitly, how it generates each element of the group.


Week of March 29 - April 2, 2010
This week we'll wrap up our proof that one cannot trisect the angle with straightedge and compass. We'll also begin studying splitting fields as a first step towards the Fundamental Theorem of Galois Theory.
The problems below form Homework #9 and are all due Monday, April 5. This puts us back on our Monday-Monday homework schedule. There will be a problem session Friday.
Monday:
  • Study: Geometric constructions - Section 13.3 and notes from class.

  • Do: no new problems
Wednesday:
  • Study: Section 13.4.
  • Do: (The examples in Section 13.4 are useful to consider for this assignment and the one that will be posted on Monday.)
    Problem 1: Consider the polynomial x4-5x2+6 in the polynomial ring over the rationals. What is its splitting field? What is the degree of this splitting field over the rationals?
    Problem 2: Find the splitting field of the element x3+x+1 of the ring (Z/2Z)[x].
Friday:
  • Study: Section 13.4

  • Do: No new problems.


Week of March 5 - 12, 2010
On Monday this week we examined the first number to be proven transcendental. On Wednesday and Friday we'll define the set of complex numbers constructible from some finite set of complex numbers. We will confirm that this set is in fact a subfield of C with some nice properties.
The problems below form Homework #8 and are all due Wednesday, March 31. Homework #9 will be assigned that Wednesday and due the following Monday to get us back on the Monday-Monday homework schedule.
Monday:
  • Study: Midterm exam day

  • Do:

Wednesday:
  • Study: Section 13.3 provides another perspective on what we're doing in class.
  • Do:
    problems

Friday:
  • Study: Section 13.3 as needed.

  • Do: nothing new today


Week of March 5 - 12, 2010
This week we will wrap up our work on algebraic extensions. Class will meet Monday as usual and Tuesday from 3:30-4:30. Wednesday's class is cancelled - please do considering using this time to attend part of the Gender Studies Symposium. The midterm exam will be distributed Friday and due Tuesday next week. Friday and next Monday's class will be on a bonus topic in algebra. On Wednesday next week we'll begin our study of classical straightedge and compass constructions.
The problems below form Homework #7 and are all due Monday, March 15. Note that this is a short assignment, I recommend completing it before the midterm is distributed on Friday.
Monday:
  • Study: Section 13.1

  • Do:
    problems
Wednesday:
  • Study: Section 13.2
  • Do:

    No new problems today.
Friday:
  • Study: Nothing new today.

  • Do:


Week of March 1 - 5, 2010
This week we focus on simple extensions, algebraic extensions and finite extensions. How do all of these types of field extensions relate to each other? We will work with algebraic elements of an extension field and their minimal polynomials.
The problems below form Homework #6 and are all due Monday, March 8.
Monday:
  • Study: Section 13.1

  • Do:
    problems
Wednesday:
  • Study: Section 13.2
  • Do:

    problems
Friday:
  • Study: Section 13.2

  • Do:


Week of February 22 - 26, 2010
We review Kronecker's Theorem on Monday and discuss further how to approach K/F in this theorem as a vector space. On Wednesday we'll define simple and algebraic extensions. Note that we should schedule the take home midterm this week.
The homework schedule has been shifted.
Monday:
  • Study: Section 13.1

  • Do: (Extension from last week.)
Wednesday:
  • Study:
  • Do:
Friday:
  • Study:

  • Do:


Week of February 15 - 19, 2010
On Monday we'll prove Eisenstein's Criterion for irreducbility and do some examples. Wednesday will be spent proving a few other convenient polynomial ring facts. By the end of the week we'll begin our climb to the Fundamental Theorem of Galois Theory. The first step is to examine field extensions.
The problems below form Homework #5 and are all due Wednesday, February 24.
Monday:
  • Study: Section 9.4, 9.5

  • Do:
    A. Show that a primitive polynomial with integer coefficients is irreducible over the integers if and only if it is irreducible over the rationals. Give an example of a non-primitive polynomial which is irreducible over the rationals but reducible over the integers.
    B. Determine which of the following polynomials are reducible over the rationals: x4+x+1, 5/2x5+9/2x4+15x3+3/7x2+6x+3/14.
    C. Section 9.4 Problem #20abc.
Wednesday:
  • Study:Sections 9.5 and 13.1
  • Do:
    D. Problem #8 in Section 9.4.
    E. Let K/F be a field extension. Prove that [K:F]=1 if and only if K=F.
    F. Suppose F is a field, and R is an integral domain that contains F as a subring. If R, considered as a vector space over F, is finite dimensional then show R is a field.
Friday:
  • Study:

  • Do:


Week of February 8 - 12, 2010
We'll start off this week examining how factoring polynomials over an integral domain relates to factoring polynomials over that domain's field of fractions. This will bring us to two versions of Gauss's Lemma. We'll then examine various tests of polynomial reducibility and irreducibility.
The problems below form Homework #4 and are all due Wednesday, February 17.
Monday:
  • Study: Section 9.3

  • Do:
    A. Suppose R is a UFD and (p) is a prime ideal in R. Also suppose f(x) is a polynomial in R[x] all of whose coefficients are divisible by p. Consider the mapping from R[x] to (R/(p))[x] that reduces the coefficients of a polynomial mod (p). First make sure you understand this mapping and believe that it is a homomorphism. (No need to write anything up for that - we did part of the proof in class on Feb. 1st.) Give a proof that this mapping takes f(x) to 0+(p) in (R/(p))[x]. This was part of our proof of Gauss's Lemma.
    B. Prove the Lemma from class today that directly followed our proof of Gauss's Lemma (and directly preceded our proof of Prop 9.3.5).
Wednesday:
  • Study: Section 9.3
  • Do: C. Is f(x)=21x3-3x2+2x+8 irreducible over the rationals?
    D. Is f(x)=x5+2x+4 irreducible over the rationals?
    E. Section 9.3 Problem #2.
Friday:
  • Study: Sections 9.3, 9.4

  • Do: no new problems today


Week of February 1 - 5, 2010
This week we will begin our in-depth study of polynomial rings. We'll first look at how ideals, polynomial rings and quotients mix together. Then we'll sort out that as claimed a polynomial ring over a field is a Euclidean domain. Next week (and perhaps Friday) we will look into factoring polynomials in R[x] and how that process relates to the properties of the base ring R.
The problems below form Homework #3 and are all due Wednesday, February 10.
Monday:
  • Study: Sections 9.1 and 9.2

  • Do:
    A. Show that the polynomial 2x+1 in (Z/4Z)[x] has a multiplicative inverse in (Z/4Z)[x].
    B. Let p be a prime integer. Are there any nonconstant polynomials in (Z/pZ)[x] that have multiplicative inverses? Explain your answer.
    C. Prove that the ideal (x) in Z[x] is prime but not maximal.
    D. Prove that the ideal (x) in Q[x] is maximal.
Wednesday:
  • Study: Section 9.2

  • Do:
    E. Section 9.2 Problem #3.
    F. Section 9.2 Problem #4.
    G. Section 9.3 Problem #6abc. Skip the part about alpha but do determine the characteristics of these rings.
Friday:
  • Study:

  • Do:


Week of January 25 - 29, 2010
This week we will construct the field of fractions of an integral domain. We will then check in briefly with Chapter 8 to define Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains. After that we begin our first main topic of the semester - polynomial rings.
The problems below form Homework #2 and are all due Wednesday, February 3.
Monday:
  • Study: Section 7.4, 7.5

  • Do:
    A. Show that the operation of addition in the field of fractions Q of a division ring R is well defined.
    B. Let Z[i]={a+bi : a,b are in Z}. Show that the field of fractions of Z[i] is ring-isomorphic to Q[i]={r+si : r,s are in Q}. Note that Z denotes the integers and Q denotes the rationals.
    C. Let F be a field. Show that the field of fractions of F is isomorphic to F.
    D. Let D be an integral domain and let F be its field of fractions. Show that if E is any field that contains D, then E contains a subfield isomorphic to F. (Thus, the field of fractions of an integral domain is the smallest field containing that integral domain.)
Wednesday:
  • Study: Chapter 8 (selectively - we are skipping quite a bit of the material)

  • Do:
    E. Problem #5ab in Section 8.2.
    F. Read the definitions of irreducible and prime ring elements on p.284. Show that in an integral domain all prime elements must be irreducible.
    G. Show that in a PID, a nonzero element is prime if and only if it is irreducible.
Friday:
  • Study: Chapter 8 as needed, Chapter 9.1

  • Do: no new problems


Week of January 18 - 22, 2010
This week we will study the properties of ideals, and begin to construct the field of fractions of a ring. Next week we'll finish this construction. We will then check in briefly with Chapter 8 to define Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains.
The problems below form Homework #1 and are all due Wednesday, January 27.
Monday:
  • Study: no class this day

  • Do:
Wednesday:
  • Study: Section 7.4 and Appendix 1 on Zorn's Lemma (p. 907)

  • Do: Page 910 Problems #1 and #2. Please skip the part of each of these problems that refers to well-orderings.
Friday:
  • Study: 7.4

  • Do:
    A. Show that in a commutative ring with unity the ideal generated by a single element (a) is equal to {ra : r is in the ring}. (We used this fact on the first day of class to prove Proposition 7.4.9(2).)
    B. Problem #5 in Section 7.4.
    C. Problem #7 in Section 7.4. To do this problem you will need to refer to the reading in Section 7.4 for relevant definitions and results. You are welcome to use Prop 13 and Cor 14 even though we did not discuss them in class.