Math 4108: Abstract Algebra II Welcome to Math 4108: Abstract Algebra II with Prof. Heitsch!
 Times & Places Lectures are in Skiles 243 on Mondays and Wednesdays from 4:35 -- 5:55. Office hours are currently scheduled for Tuesday from 1 - 2pm and Monday, Wednesday from 6 -- 6:30pm in Skiles 226. (Meeting at other times is possible by appointment.) Announcements Handout from Monday, January 5th: Course Syllabus. Homework Assignment 1 due Wednesday, January 14th. Assignment 2 due Wednesday, January 28th. Assignment 3 due Wednesday, February 11th. Assignment 4 due Wednesday, March 11th. Note that in the first problem "at least" should be "at most." Assignment 5 due Wednesday, April 1st (no joke!). Practice Problems Prove that a semigroup G is a group if and only if it contains right inverses and right identity. Try left inverse and right identity instead and figure out why that's a problem. Think about why a semigroup G is a group if and only if for all elements a, b in G, the equations ax = b and ya = b have solutions in G. What is a group presentation for Sn (the symmetric group with n elements)? Start with S3. Describe all abelian groups of orders 2^6, 11^6, 7^5, 2^4 3^4, 2^3 3^4 5. _Abstract Algebra_ Section 2.10 # 1 - 3. Find cl(a), C(a) for all elements a in the groups S_3, S_4, D_8, and the multiplicative group of quaternion units. Verify the class equation for these groups. Generalize D_8 to D_2n for cl(a), C(a), and the class equation. Prove that conjugacy is an equivalence relation. Prove that the centralizer of an element is a subgroup. Find an element a and group G such that C(a) is not normal. _Abstract Algebra_ Sec. 6.1 #1, 2, 3, 6, 8, 9, 12. _Abstract Algebra_ Sec. 4.5 #1, 2, 3, 5, 6, 7, 8, 10, 11, 12. Complete the proofs (big phi is a homomorphisms, operations on F are well-defined, the set of 'fractions' F is a field) from class. _Abstract Algebra_ Sec. 4.6 #1, 2, 3, 5, 6, 7, 8, 10. _Abstract Algebra_ Sec. 4.7 #1, 2, 3. _Abstract Algebra_ Sec. 5.2 #3, 5, 7, 8, 10, 11, 12. _Abstract Algebra_ Sec. 5.3 #1, 2, 5, 7. _Abstract Algebra_ Sec. 5.4 #1, 3, 6, 7. Complete the proof of Theorem 5.6.5 in _Abstract Algebra_ by showing that the map \psi from F[x] to K gives \psi(g(x)) = g(a) for a polynomial g(x) from F[x] and a = \psi(x) = x + M in the field K = F[x]/M for M = (p(x)). Let K be a field of characteristic p, a prime. Prove that K contains a subfield isomorphic to Z_p. _Abstract Algebra_ Sec. 5.5 #2, 3, 4, 5, 6, 7. Check that q(x) is the minimal polynomial over Z_p for the element a = x + (q(x)) in the field A = Z_p[x]/(q(x)). _Abstract Algebra_ Sec. 6.2 #1 - 5, 7 - 9. _Abstract Algebra_ Sec. 6.3 #1, 2, 3. Check that \tau**[g(x) + (f(x)] = g'(t) (f'(t)) is an isomorphism of F[x]/(f(x)) onto F'[t]/(f'(t)) with the required properties. Convince yourself that M = {f(x) \in F[x] | f(v) = 0} is an ideal of F[x]. Confirm that the mapping \sigma(v) = \theta* \tau** (\phi**)^{-1} has the required properties. Convince yourself that E and E' are also splitting fields for f(x) and f'(t) over F(v) and F'(w), respectively. _Topics in Algebra_ Sec. 5.5 #1, 2, 3, 4. _Topics in Algebra_ Sec. 5.6 #1 - 6. Prove that the fixed field of G is subfield of K. _Topics in Algebra_ Sec. 5.6 #7 - 11. Prove that the elements of S_n acting appropriately on F(x_1,...,x_n) are automorphims of F(x_1,...,x_n). Prove that each \alpha_i is invariant under the elements of H. Show that if T is normal, then p(x), the minimal polynomial of a, where T = F(a), has all its roots in T. _Topics in Algebra_ Sec. 5.7 #1 - 5. _Topics in Algebra_ Sec. 2.7 #4, 5. Links Check your grades on T-Square. Student are expected to follow the Georgia Tech Academic Honor Code and Student Code of Conduct.
C. E. Heitsch
Spring semester 2009