Math 536 Homework 4
Due: Friday, February 8
1. Let : B C be a surjective homomorphism of groups. Show that if F is free abelian
and : F C is any homomorphism, then there exists a homomorphism : F B
making the diagram below commute, i.e. such t
Math 536 Homework 12
Due: Friday, April 26
1. It is a fact that, if p is prime, then Sp is generated by a transposition and a p-cycle.
(a) Show that if a polynomial f (x) Q[x] is irreducible of prime degree p and has
exactly p 2 real roots, th
Math 536 Homework 10
Due: Friday, April 12
1. Let R be a domain containing a eld k as a subring. Suppose that R is a nite
dimensional vector space over k under the ring multiplication. Show that R is a eld.
2. Let F be a eld of characteristic
Math 536 Homework 6
Due: Friday, March 1
1. Let G be a simple group of order 60. Show that G A5 . (Hint: Consider the dierent
possibilities for the number of Sylow 2-subgroups.)
2. Let p, q be distinct primes. Prove that any group of order p
Math 536 Homework 1
Due: Friday, January 18
1. Let G be a group, and let H1 , . . . , Hk be subgroups of G. We say that G is an internal
direct product of the subgroups Hi if the map
(h1 , . . . , hk ) h1 h2 hk : H1 H2 Hk G
is an isomorphism o
Math 536 Homework 9
Due: Wednesday, March 27
1. Suppose that R is a commutative ring with identity such that every submodule of every
free R-module is free. Show that R is a PID.
2. Let R := Z[X ]. Give an example of a nitely generated R-modul
Math 536 Homework 11
Due: Friday, April 19
1. Give explicit generators for the subelds of C which are splitting elds of the following
polynomials over Q, and nd the degree of each such splitting eld.
(a) X 3 2
(b) (X 3 2)(X 2 2)
(c) X 2 + X +
Math 536 Homework 7
Due: Friday, March 15
1. Let D be an integer 1 and let R be the set of all elements a + b D with a, b Z.
(a) Show that R is a ring.
(b) Let N : R Z be the norm map, i.e. the map given by
N (a + b D) = (a + b D)(a b D).
Math 536 Homework 5
Due: Friday, February 22
1. Let G and H be nite groups of relatively prime order. Show that Aut(G H ) is
isomorphic to Aut(G) Aut(H ).
2. For which primes p and positive integers n is every p-Sylow subgroup of the symmetric
Math 536 Homework 2
Due: Friday, January 25
1. Show that a group of order 2m, m odd, contains a subgroup of index 2. (Hint: You
may use Cayleys theorem, Corollary 4, page 120 in Dummit and Foote, or Jacobson,
Corollary on page 38.)
2. Let K be
Math 536 Homework 3
Due: Friday, February 1
1. Let G be the group of invertible 4 4 matrices over the complex numbers, and let M
be the set of all 4 4 complex matrices.
(a) Consider the action of G G on M given by (g, h) acts on m by the matri
Math 536 Homework 8
Due: Friday, March 22
1. (a) (Euclids algorithm for nding the gcd.) Let a1 , a2 be nonzero elements of a
Euclidean domain R. Dene ai and qi recursively by a1 = q1 a2 + a3 , ai =
qi ai+1 + ai+2 where (ai+2 ) < (ai+1 ). Show