Math 401

Fall 2014

A first course in rings and groups out of Shifrin.

• Syllabus.
• Errata for the textbook.
• Homework 1 (due 9/2):
  • Section 1.1: #4 (choose four parts), 7, 8a, 16 (optional).
  • Section 1.2: #1 (choose three parts), 6, 7, and the following variation on #11: Suppose that k is not a square. Show that there are no integers m and n such that m² = k n². Hint: consider the prime factorization of m and n. Maybe warm up with k = 2 and k = a prime.
  • Solutions: pdf, tex.
• Homework 2 (due 9/11):
  • Section 1.3: #14, 15, 16 (optional), 20 (choose three parts), 21 (choose three parts), 25, 38c. Read through all the problems in the section.
  • Solutions: pdf, tex.
• Homework 3 (due 9/18):
  • Section 1.4: #3, 8, 11, 19bd, 20, and the following variation on #7: Suppose that R is any ring, c, x, y ∈ R, and c is neither zero nor a zero-divisor. Prove that if cx = cy then x = y.
  • Section 2.1: #8, 15, and show that the sets R appearing in #12 and 13 are indeed rings, i.e. they contain 1 and are closed under addition, subtraction, and multiplication.
  • Section 2.2: #5, 6ce.
  • Solutions: pdf, zip.
• Handout on Writing by Jack Lee.
• Homework 4 (due 9/25):
  • Section 2.3: #6, 9 (choose three parts), 10, 17.
  • Section 2.4: #1 (choose two parts), 3, 6 (but don’t worry about Δ and Proposition 4.4), 7 (choose either part), 10, 11 (optional).
  • Solutions: pdf, tex.
• What is the smallest prime?
• Homework 5 (due 10/2):
  • Section 3.1: #1 (choose one of c,d,e), 6, 10 (all parts), 13, 14, 19, 20 (choose one of b,c).
  • Solutions: pdf, tex.
• Homework 6 (due 10/16):
  • Section 3.2: #6 (choose two parts), 7, 10, 14.
  • Section 3.3: #2 (choose three parts), 3 (choose one part), 7, 10.
  • Solutions: pdf, tex.
• Midterm 1, Solutions, tex.
• Homework 7 (due 10/23):
  • Section 4.1: #4 (choose 3 parts), 10, 12, 15 (all parts), 16, 17a, and the following variation on 17b: In Z, let I = (4) and J = (10). Identify I + J and I ∩ J.
  • Beware that there are some printing errors in #16 – see the errata linked above.
  • Optional: In class we saw that Z₂[x]/(x²+x+1) is a field with 4 elements. Similarly one can show that Z₂[y]/(y³+y+1) and Z₂[z]/(z³+z²+1) are fields with 8 elements. Show that they are isomorphic.
  • Solutions: pdf, tex.
• Homework 8 (due 10/30):
  • Section 4.2: #1, 5, 11abc, 13, 22.
  • Section 4.3: #1 (choose one part), 2 (choose three parts), 10 (choose two parts), 15.
  • Solutions: pdf, tex.
• Homework 9 (due 11/6):
  • Section 5.1: #3, 4, 11 (just give the answers, no proofs), 20.
  • Section 5.2: #4 (he's referring to (1), (2), and (3) on page 159), 5.
• Midterm 2, Solutions, tex.
• Homework 10 (due 11/13):
  • Make a tetrahedron and a cube out of paper. Determine the size of their symmetry groups.
  • Solution: For the tetrahedron the answer is 24, or 12 if you take only rotations. For the cube it is 48, or 24 if you take only rotations.
• Homework 11 (due 11/20):
  • Section 6.1: #1, 4, 5, 17, 18 (choose four parts).
  • Section 6.2: #6b, 8, 12, and the following variation on #7: show that GL₂(Z₂) ≅ S₃.
  • Solutions: pdf, tex.
• Homework 12 (due 12/2):
  • Section 6.3: #4, 5, 6, 12, 15, 23, 25, 31 (optional), 33.
  • Solutions: pdf, tex.
• Homework 13 (not due):
  • Section 6.4: #1, 5.
  • Section 7.1: #3, 10. For #10, use the presentation Sq = { 1, r, r², r³, s, sr, sr², sr³ } with the relation rⁿs = sr^4-n.
    To say that a group acts on itself by conjugation means that we define g⋅h = ghg^-1.
  • Section 7.2: #1.
  • Solutions: pdf, tex.
• Final Exam, Solutions, tex.