# Math 415, Modern Algebra I, Fall 2006Harold P. Boas

## What's new

Saturday, December 9
The class turned in a great performance on the final exam! There was one score of 100, and the average was higher than on either the second exam or the third exam.
To give recognition to those of you who improved on the final exam, I modified the algorithm for computing the course average. Instead of simply averaging your five scores (three exams, homework, and final exam), I first replaced the lowest of your five scores by your final exam score and then averaged the five numbers. (If your final exam score is the lowest of the five scores, then your course average did not change; otherwise, your course average increased.)
You can check your grade at WebCT. The course average labelled "preliminary" is the average you had under the original system; the average labelled "final" is the new system.
Good luck with the rest of your exams, and have a safe winter break!
Friday, December 8
The final examination was given. Solutions are available.
Tuesday, December 5
We reviewed for the comprehensive final examination to be given on Friday, December 8, from 12:30 to 2:30pm. I will have office hours every day this week from 11:00am to noon.
Thursday, November 30
The third examination was given. Solutions are available.
Tuesday, November 28
We reviewed for the third examination to be given on Thursday, November 30.
Tuesday, November 21
We discussed the division algorithm and unique factorization in polynomial rings along with the Eisenstein irreducibility criterion. The assignment (not to hand in) from Exercises 23 on pages 218-220 is exercises 1, 7, 11, 21, 27, and 34. Travel safely over the Thanksgiving holiday. Reminder: the third examination is on Thursday, November 30.
Thursday, November 16
We discussed the problem of finding zeroes of polynomials over a field. The assignment for next class is to do exercises 2, 6, 14, 16, and 24 in Exercises 22 on pages 207-209.
Tuesday, November 14
We discussed the construction of a field of quotients of an integral domain. The assignment for next class is to do exercises 2, 5, and 6 in Exercises 21 on pages 196-197.
Thursday, November 9
We discussed Fermat's little theorem, Euler's generalization, and the solution of linear congruences. The assignment for next class is to do exercises 4, 6, 12, 14, and 29 in Exercises 20 on pages 189-190.
Tuesday, November 7
We discussed the notions of zero divisors, integral domains, and characteristic of a ring. The assignment for next class is to do exercises 4, 10, and 23 in Exercises 19 on pages 182-184.
Thursday, November 2
We discussed the notions of rings and fields. The assignment for next class is to do exercises 10, 18, 32, 38, and 44 in Exercises 18 on pages 174-177.
Tuesday, October 31
The second examination was given. Solutions are available.
Thursday, October 26
We reviewed for the examination by working on the true/false questions on pages 152, 142, 134, and 112.
Tuesday, October 24
We continued the discussion of normal subgroups and of factor groups, and we looked at the notions of simple group and the commutator subgroup. The assignment for next class (not to hand in) is to do exercises 5, 13, 29, and 31 in Exercises 15 on pages 151-154. Reminder: the second examination is scheduled for Tuesday, October 31.
Thursday, October 19
We discussed different characterizations of normal subgroups, and we looked at the group law defined on the cosets of a normal subgroup. The assignment for next class is to do exercises 6, 14, 24, 25, and 27 in Exercises 14 on pages 142-144.
Tuesday, October 17
We discussed the concepts of group homomorphism, kernel, image, inverse image, and normal subgroup. The assignment for next class is to do exercises 18, 28, 36, and 47 in Exercises 13 on pages 133-135.
Thursday, October 12
We discussed direct products of groups and the fundamental theorem of finite abelian groups. The assignment for next class is to do exercises 8, 24, 39, 40, and 46 in Exercises 11 on pages 110-111.
Tuesday, October 10
We discussed cosets and Lagrange's theorem. The assignment for next class is to do exercises 4, 12, 20, and 26 in Exercises 10 on pages 101-104.
Thursday, October 5
We discussed the notions of orbit, cycle, transposition, and the alternating group. The assignment for next class is to do exercises 10, 16, 34, and 36 in Exercises 9 on pages 94-96; also determine which elements of the dihedral group D4 are even permutations and which are odd permutations (you may wish to refer to the group table on page 80 of the textbook).
Tuesday, October 3
We discussed permutation groups, and in particular the examples of the symmetries of an equilateral triangle and the symmetries of a square. The assignment for next class is to do exercises 8, 12, 16, and 32 in Exercises 8 on pages 83-87.
Saturday, September 30
The class did a great job on the first examination! There were three scores of 100, and 19 of the 23 papers had scores of 80 or above. You can check your grade at WebCT.
Thursday, September 28
The first examination was given. Solutions are available.
Tuesday, September 26
Several homework assignments were returned, and we did an abbreviated review for the first examination.
Sunday, September 24
I have posted some suggestions about how you might review for the first examination (to be held on Thursday, September 28).
Thursday, September 21
We continued the discussion of subgroups of cyclic groups and the notion of the greatest common divisor. Also we touched on the idea of a group generated by more than one element.
The first examination will be on Thursday, September 28 on sections 0-7. I will be out of town that day (I am giving a lecture at Amherst College on September 27), so Dr. Geller will give the examination. On Tuesday, September 26, we will have an abbreviated class meeting (since I leave for the airport at 10:00). I expect to be available for questions in my office most of the day on Friday, September 22 (until a seminar at 15:00) and on Monday, September 25.
There is no homework to hand in next class. I suggest that in preparation for the examination, you make for yourself lists of the main definitions and theorems, and a list of examples of groups that we have encountered so far.
Tuesday, September 19
We discussed the classification of cyclic groups and the structure of subgroups of finite cyclic groups. The assignment for next class is to do exercises 6, 14, 18, and 24 in Exercises 6 on page 66.
Thursday, September 14
We discussed the concept of a subgroup of a group and, in particular, the notion of a cyclic subgroup. The assignment for next class is to do exercises 16, 30, 40, 52, and 53 in Exercises 5 on pages 55-59.
Tuesday, September 12
We discussed the definition of a group and some basic properties of groups, such as the cancellation law, the solvability of linear equations, and the uniqueness of the identity element and of inverse elements. The assignment for next class is to do exercises 6, 18, 19, and 32 in Exercises 4 on pages 45-49.
Thursday, September 7
We discussed the concepts of homomorphism and isomorphism of binary structures. The assignment for next class is to do exercises 6, 12, 18, 26, and 33 in Exercises 3 (pages 34-36).
Tuesday, September 5
We discussed the definition and some examples of binary operations, and we worked on some exercises at the end of Section 2. The assignment for next class is to do exercises 6, 26, 28, and 36 in Exercises 2 (pages 25-28). A couple of these exercises call for proofs: remember that a proof should be written in sentences.
Thursday, August 31
We discussed some group structures related to circles: modular arithmetic and multiplication of complex numbers of modulus 1. The assignment for next class is to do exercises 20, 22, 28, 30, and 36 in Exercises 1 (page 19).
Tuesday, August 29
At the first class meeting, we discussed relations and equivalence relations, partitions, and cardinality (topics from Section 0). The standing assignment is to read the textbook. Also, for next class do exercises 9, 15, 25, 29, and 31 in Exercises 0 (pages 8-10).
Monday, August 28
This site went live today. Watch for regular updates. The first-day handout is available online.