Modern Algebra I

  • Syllabus
  • Class Calendar: This is a calendar of what we actually cover, the calendar of what we plan to cover is on the syllabus.
    1. Introduction to Groups. (8/29/2016)
    2. Reviewed groups introducing the integers mod n, dihedral groups, and symmetric groups.  We looked at subgroups and finished with a proof of Lagrange’s Theorem. (9/12/2016)
    3. We discussed subgroups, normal subgroups, factor groups, homomorphisms, and the First Homomorphism Theorem. Next week will do the other homomorphism theorems and Cauchy’s Theorem. (9/19/2016)
    4. We reviewed normal subgroups and factor groups and then briefly discussed the homomorphism theorems, IOU the proof of the second homomorphism theorem next week.  Then we went through Cauchy’s theorem and covered some number theory. Next week we will cover a couple more number theory facts so you have examples of induction, we will review Cauchy’s theorem and its consequences, and time allowing look at direct products and finite abelian groups.
    5. We reviewed Cauchy’s Theorem, proved the Second Homomorphism theorem, covered a little more number theory, and introduced the ideas of internal and external direct products (section 9) in preparation for discussing the Fundamental Theorem of Finite Abelian Groups.  Next week we will look at the proof of the the Fundamental Theorem of Finite Abelian Groups and at symmetric groups.
    6. We went over the Fundamental Theorem of Finite Abelian Groups and discussed symmetric groups.  next week is the exam on groups, you will have the full time for the exam, there will be no lecture. (10/11/2016)
    7. Group Theory Exam on 10/17/2016 (Study Guide)
    8. We covered the most of the material from 4.1 and 4.2, you should be able to start looking at exercises from those sections.  We also covered some from 4.3 and 4.4.  Next week we will finish off 4.3 and 4.4 and introduce 4.5. (10/24/2016)
    9. Finished 4.3 and 4.4 and discussed the direction we are headed in.  (10/31/2016)
    10. We covered material from 4.5 which is polynomials over fields in general, and started 4.6.  We will finish 4.6 next Monday and the exam will be the Monday after that.  (11/7/2016)
    11. We finished 4.6 and looked at exercises 6 through 11 which together showed that:
      An automorphism \[\phi:F[x]\rightarrow F[x]\] fixes all the elements of the field \(F\), and so preserves irreducibility, if and only if \[\exists b,c\in F, b\not=0, \forall f(x)\in F[x], \phi(f(x))=f(bx+c).\]
      Next class we will have the exam on unit 2, I will post a study guide soon.  For now look through the exercises listed below, study the statements of the major theorems we have covered, and study your definitions.
    12. Unit 2 Exam , Ring Venn Diagram
    13. We looked at field extensions and their relation to roots of polynomials and vector spaces.  Next week we will discuss either constructible numbers or transcendental numbers or possibly roots of polynomials and the Fundamental Theorem of Algebra.
    14. We went through a proof of the Fundamental Theorem of Algebra and discussed why some numbers are constructible and others are not.
    15. Make-Up Exam 12/12/2016 at 6pm
  • Assignments:
    • Unit 2: Do one problem from each section and then do two additional problems of your choice for a total of seven problems.  You may do two additional problems for extra credit. Due 11/21/2016
      • p.133 – #’s 2, 3, 11, 19;
      • p. 139 – #’s 2, 3, 8;
      • p. 146 – #’s 1, 2, 5, 9, 21, 22 (for 21 and 22 use 20);
      • p. 150 – #’s 1 (remember \(a^2+b^2 = (a+bi)(a-bi)\) ), 2, 8;
      • p. 163 – #’s 2, 6, 11, 12, 16 (try 13 and 14 first); p. 171 – #’s 3, 5 (look at example 5)
    • Unit 1: Do one 5pt, one 10pt, and one 15pt problem, then do at least 20 more points worth of problems for a total of at least 50pts.  Problems combined with an “&” must both be done in order to count. These are due on 10/24/2016:
      • 5 pt Problems: p.46 – #’s 8,9; p.50 – #’s 5; p.54 – #’s 6, 19; p.73 – #’s 7, 14, 15, 20; p.82 – #’s 3, 12; p.91 – # 2; p.96 – # 1; p.117 – #’s 6, 7,9,10,13,14; p.123 – # 5
      • 10 pt Problems: p.46 – #’s 13, 24;  p.54 – #’s 14&16, 30; p.63 – #’s 8, 32; p.73 – #’s 23, 26, 35; p.87 – # 4; p.91 – #’s 5, 8; p.96 – #’s 2, 3; p.101 – #’s 1, 2, 3; p.117 – # 5,11,12,15; p.123 – 2,3,4
      • 15 pt Problems: p.46 – #’s 26 & 27, 28; p.50 – #’s 3 & 4; p.63 – #’s 36; p.73 – #’s 29, 34; p.82 – # 10; p.96 – #’s 5, 6 
  • Links & Handouts: