Homework for Math511A

08/27: Read the examples after the definition of a group, especially you should understand the dihedral group D_n. Read the Proposition 1.6 and 1.7 on cyclic groups. Do Exercises 1.5 and 1.6 (hand in just this part).

08/29: Exercise 1.7 (make sure you know what these groups are!); check that Inn(G) is a normal subgroup of Aut(G).

09/03: Exercises 2.1, 2.3, 2.4

09/05: Exercises 3.2, 3.3. In addition, show that there is a unique normal subgroup K of order four in S_4 and S_4/K is isomorphic to S_3.

09/10: Exercise 4.2.

09/12: Show that if G is of order 24 and the center of G is trivial, then G is isomorphic to S_4. Hint: show first that there are four 3-Sylow subgroups.

09/17: (1) Understand the presentation of the dihedral group D_n. (2) p.45, Exercise 52. Note: the group Q_2 is our Q_8 discussed in the class today.

09/19: Classify all abelian groups of order 1500.

09/26: (1) Check two definitions from the class of the Hamilton quaternion are the same. (2) Prove that a commutative ring (with 1) is a field if and only if the only the only ideals are (0) and R itself. (3) Prove that a finite integral domain is a field.

10/01: Chapter 2, Exercises 2.2 and 2.3.

10/03: Chapter 2, Exercise 3.4.

10/08: Chapter 2, Exercise 5.1.  Section 8, problem 24.

10/10: Chapter 2, Section 8, problem 30. Also think about this problem (this is a bit hard, it's fine if you don't get it): an integral domain is a UFD if and only if every nonzero prime ideal contains a prime element.

10/15: (1) Show that Q[x]/(x^3 - 3x -3) is a field and simplify (1+x+x^2)^{-1}. (2) Show that C[x, y]/(y^2 - x^3) is an integral domain. Let K be its field of fractions. Show that it is isomorphic to C(x). Do the same for the polynomial y^2 - x^3 - 1, but this time show that the field of fraction is not isomorphic to C(x).

10/17: (1) Find the splitting field of x^9-1 over Q. (2) Chapter 3, Exercise 1.1. The notation A in the book is the Qbar in our class, i.e. the algebraic closure of Q in C.

10/22： (1) Find the splitting field of x^8 - 1 over Q. Determine the Galois group of it. (2) What is Galois group of Q(e^{2 pi sqrt{-1} / n}) over Q? Is Q(e^{2 pi sqrt{-1} / n}) the splitting field of any polynomial over Q?

10/24: (1) Show that over a finite field of characteristic p, an irreducible polynomial f(x) divides x^{p^n}-x if and only if deg f(x) divides n. (2) How many degree n irreducible polynomials are there over F_{p^n}? (this one is a little hard, you can skip it if you prefer).

10/31 and 11/5：(1) Show that if E/F is a normal and K/E is also normal then E/F is normal. Show also that if E/F and K/F are normal then EK/F is normal. (2) If E/F is algebraic and F is perfect, then E is perfect. Does the converse hold? Does the converse hold if E/F is finite? (3) Let E = F_p(x, y), F = F_p(x^p, y^p). Show that E/F is finite but not separable. Show also that E/F is not monogenic. (4) Show that degree two extensions are all normal. Determine the possible Galois groups. (5) Consider Q(u) where u satisfies u^3 + u^2 - 2u - 1 = 0. Show that this is normal extension of Q of degree three. What is the Galois group?

11/7: Find the splitting field of x^4 -2 over Q. Find the Galois group and all its subgroups. Which of them are normal? Using Galois theory, find all intermediate extensions. Which of them are Galois over Q?

11/12: Find degree 3 and 5 Galois extensions of Q. Can you find infinitely many of them?

11/14: (1) Find the minimal polynomial of (cubic root of 2) + sqrt{3} over Q? (Hint: use the method in the proof we did in class) (2) Assume that E/F finite Galois. Show that an intermediate extension M/F is Galois if and only if for all \sigma \in Gal(E/F), \sigma(M) = M.