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people:fer:401ws:fall2018:homework

Math 401 - 01 Homework (Fall 2018)


Problem Set 13 (complete) Due: 12/10/2018, optional (bring to review session)

  1. Let D be an I.D., D[x] the ring of polynomials over D, and D(x) the field of rational functions over D. Let F be the field of fractions of D, F[x] the ring of polynomials over F, and F(x) the field of rational functions over F. Show that D(x)=F(x).
  2. Prove that the operation that defines the external semi-direct product is in fact associative.
  3. Prove that the two non-abelian semi-direct products of C7 with C3 are isomorphic. (Hint: use the homomorphism discussed in class, given by: au,bv1)

Problem Set 12 (complete) Due: 12/03/2018. Board presentation: 12/07/2018

  1. Chapter 14, problems 12, 14. Warning: pay attention to the definition of AB.
  2. Chapter 14, problem 28. What about the converse?
  3. Write nZ as md0, where d0 is the last digit (base 10) and m consists of all other digits. In other words, n=10m+d0. Prove that n is divisible by 7 iff m2d0 is divisible by 7.

Problem Set 11 (complete) Due: 11/20/2018. Board presentation: 11/27/2018

  1. Chapter 12, problem 18. Moreover, if R is commutative, then S is an ideal of R.
  2. Chapter 12, problem 28.
  3. Chapter 13, problem 52.
  4. Chapter 13, problem 34.

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people/fer/401ws/fall2018/homework.txt · Last modified: 2018/12/07 14:11 by fer