ALGEBRA PRELIMINARY EXAM JANUARY 18, 2000

1. Show that a Sylow p-subgroup of D_{2n}, the dihedral group of order 2n, is cyclic and normal for every odd prime p.

2. Show that every automorphism of S_{3}, the symmetric group of order 6, is inner, that is, is conjugation by an element of S_{3}.

3. Determine all abelian groups of order 144 up to isomorphism.

4. Let f(x) = x^{8} − 16. Find the Galois group of the splitting field of f(x) over the field of :

(a) rational numbers ;

(b) real numbers ;

(c) integers modulo 17.

5. Find two non isomorphic rings of 9 elements whose additive groups are isomorphic. You must show that these rings are not isomorphic and that their additive groups are isomorphic.

6. (a) Let A be a commutative ring. Prove that every maximal ideal of A is a prime ideal of A.

(b) Show that the ideal (3, X) of Z[X] generated by 3 and X is a maximal ideal of Z[X].

(c) Find a prime ideal of Z[X] that is not maximal.

7. Let Z[i] denote the ring of Gaussian integers. Prove or disprove: Z[i] ⊗_{Z} R ≅ C as rings.

8. (a) Prove or disprove : The Z-module Z/2Z is projective.

(b) Prove or disprove : The Z-module Z is flat.

(c) Prove or disprove : The Z-module Q is injective.