Fall-04. Math 113. Introduction to Abstract Algebra

Lectures by Alexander Givental , MWF 12-1, 4 Evans
  • Office hours: Th 4-6 p.m., at 701 Evans
  • Text: R. Solomon, Abstract Algebra , Brooks/Cole. Most likely we will cover the chapters 1-9, 10-14, 16-18.
  • Grading: Final (50%)+Midterm (25%)+Homework (25%)
  • HW: Weekly homework assignments will be posted to this web-page, and your solutions will be due on Fridays in the class (the 1-st one is due Sept. 10).
  • Reader: Typically your homework will be returned to you on the first Wednesday following the due day (with some of the problems) graded by Yun Long . All re-grading requests should be addressed not to her directly but to me.
  • Midterm: Friday, October 15; it will be based on chapters 1-7 of the textbook.
  • GSI: Alex Dugas asdugas@math.berkeley.edu will hold office hours (for all sections of Math 113) at 891 Evans on Wed 9-12, 2-4 and Thu 9-11, 3-6.
  • Some comments about the final exam (Monday, December 20, 5-8 p.m. at 70 Evans). It is going to be a closed-books/closed-notes exam. It will consist of several problems on the subjects we studied. The problems are not from the book, but could be there, and in this sense are similar to what you had on the homework. The exam is comprehensive, i.e. the problems can address any topic we studied. To remind you, we studied Ch. 1-9, 10-14, and, in the last two lectures, some part of Ch. 16-18 [More specifically, we studied the beginning of Ch. 16 (possibility of certain constructions by straightedge and compass), a little bit of Ch. 17 (essentially the theorem on p. 183) and the essense of Ch.18 (impossibility of certain constructions).]

    Solutions to homework problems (.pdf)

    HW1 (due by 09.10.04)

    1. Prove Euclid I.6 : if two angles in a tiangle are equal to each other then the triangle is isosceles.

    2. Suppose that the triangles ABC and DEF have |AB| = |DE|, |AC| = |DF| and the angle ABC equal the angle DEF. Does this imply that the triangles are congruent? (If yes prove it, if no give a counter-example.)

    3. Solve the problems 1.10, 1.18, 2.14 from the book.

    HW2 (due by 09.17.04)

    1.15, 1.16, 1.17, 2.20a, 2.26b

    HW3 (due by 09.24.04)

    1. Prove that a group homomorphism f from a group G to a group H maps the identity element of G to the identity element of H and maps inverse elements to inverse elements (compare to Exercise 3.2).

    2. Solve the problems 3.7, 3.15ab, 3.18.

    HW4 (due by 10.01.04)

    4.5bc, 5.5, 5.6, 5.9

    HW5 (due by 10.08.04)

    5.22, 5.25, 6.22, 6.24b, 6.27

    HW6 (due by 10.15.04)

    1. On the elliptic curve y^2=(x-1)(x-2)(x-3), find all points P which satisfy the condition P+P=0 with respect to the group law + on the curve. Show that these points form a subgroup isomorphic to the Klein group V_4.

    2. Use the algorithm from the proof of the Newton-Waring theorem to express r^4+s^4+t^4+u^4 as a polynomial in the elementary symmetric functions of the four variables r,s,t,u.

    3. Solve the problems 7.5abc from the book.

    HW7 (due by 10.22.04)

    7.8, 7.9, 7.10c, 7.12, 7.13

    HW8 (due by 10.29.04)

    8.1, 8.6ab, 8.7, 8.10.

    HW9 (due by 11.05.04)

    9.8cde, 9.9, 9.15

    HW10 (due by 11.12.04)

    1. Show that the rotation group of the dodecahedron is isomorphic to the group A_5 of all even permutations of 5 symbols (it is called the alternating subgroup in S_5). Hint: the rotations permute 5 cubes formed by the diagonals in the pentagonal faces of the dodecahedron (each cube having one edge in each of the 12 faces of the dodecahedron).

    2. Find the greatest common divisor of 50339 and 128243.

    3. Referring to the picture on p. 105 of the book, find the ratio of the areas of the star ACEBD and the small pentagon FGHIJ.

    4. Solve 10.13, 10.14.

    HW11 (due by 11.19.04)

    11.7d, 11.16, 11.19, 11.20ab

    HW12 (due by Wed. 11.24.04)

    Your score for this homework will be multiplied by the factor 3/2 to compensate for shorter week and possible influence of pre-holiday anxiety.

    12.6, 12.10, 12.11, 12.17b, 12.24

    HW13 (due by 12.03.04)

    1. Prove that the group of units of any finite field is cyclic.

    2. 13.13, 12.30abe

    HW14 (due by 12.10.04)

    14.18, 16.5, 16.15, 16.17, 18.14