Math 5031 - Homework 5
Due 10/07/05
1. An abelian group A is said to be a torsion group if every element of A
has nite order. Show that the additive group Q/Z is torsion, and that
it has one and only
Math 5031 - Homework 1
Due 9/9/05
1. Show that if every element a in a group G has order 2, that is, a2 = e,
then G is an abelian group.
2. Assume that the equation abc = e holds for some a, b, c in a
Math 5031 - Homework 2
Due 9/16/05
1. Semidirect products. We dene G to be a semidirect product of subgroups H and N if N is normal, G = N H and H N = cfw_e.
(a) Let G be a group and let H , N be subg
Math 5031 - Homework 8
Due 11/04/05
1. Give an example of a ring R with a prime ideal P = 0 that is not maximal.
2. Show that Z[ 5] is not a P.I.D. (Hint: Consider the elements 3, 2 5,
2 + 5. Show tha
Math 5031 - Homework 6
Due 10/14/05
1. Dene the generalized quaternion group Qm by the presentation
Qm = a, b | a2m = 1, b2 = am , ab = ba1
for m 1. Show that Qm has order 4m. Find a concrete group wi
Math 5031 - Homework 7
Due 10/21/05
1. The Euclidean algorithm. An integral domain R with 1 is called Euclidean if there is a function d : R Z, with d(r) 0 for all r R ,
such that
(a) if a, b R and a|
Math 5031 - Homework 3
Due 9/23/05
1. Let p be prime and let G be a p-group, that is, a group of order pk for
some positive integer k . Let A be a normal subgroup of G of order p.
Prove that A is cont
Math 5031 - Homework 9
Due 11/11/05
1. Given the polynomials in C[x, y]:
f1 = x2 + y 2 1, f2 = x2 y + 1, f3 = xy 1,
prove that there are p1 , p2 , p3 C[x, y] such that 1 = p1 f1 + p2 f2 + p3 f3 .
(Not
Math 5031 - Homework 11
Due 11/28/05
It will be shown in this assignment that the eld C of complex numbers is
algebraically closed. We regard C as the splitting eld of the polynomial X 2 + 1
over R, s
Math 5031 - Homework 12
Due 12/05/05
1. Splitting elds and Galois groups of polynomials. For each of the
following polynomials (i) nd the splitting eld, K , over Q, (ii) determine
the degree of the ex
Math 5031 - Homework 10
Due 11/18/05
1. Let K = Q(), where is a root of the equation
3 + 2 + + 2 = 0.
Express (2 + + 1)(2 + ) and ( 1)1 in the form a2 + b + c, with
a, b, c Q.
2. Let be an algebraic e
Math 5031 - Final
Due 12/19/05
In this exam youll prove a number of famous impossibility results. Heres
some background on geometric constructions using a compass and an (unmarked) straightedge, taken
Math 5031 - Homework 4
Due 9/30/05
1. Prove that every group of order |G| < 60 is solvable.
The following results can be freely used. (Some will have been shown in
class but possibly not all. Make sur
am
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Math 5031 - Homework 13
Due 12/12/05
Transcendence of (in ten easy steps). The goal of this assignment is
to derive the result, rst proved by Lindemann in 1882, that is transcendental
over Q. Well mak