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 one subgroup of order n, for each positive integer
n. F
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 group G.
Does it follow that bca = e? How about bac =
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 subgroups with N normal. Let
x be conjugation by an element
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 that 3 is irreducible but not prime.)
3. Show that the ide
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 with
order 4m having the same relations as Qm . (Hint: co
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|b, then d(a) d(b), and
(b) if a, b R, with b = 0, then
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 contained in the center of G.
2. Let p be a prime number. S
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 .
(Note: you do not need to nd the pi explicitly.)
2. A varie
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, so that
C = R( 1) = R[X]/(X 2 + 1).
1. Using basic facts
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 extension, [K : Q], (iii) nd the Galois group G(K : Q),
a
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 element over a eld k. Suppose that has odd
degree. Show
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 from the text by Larry Grove. Choose two points
on the
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 sure that you can prove them, although you
do not need to
am
a mgmmm . AW
HAMIM _ _ mm _{,,_,_,____,,._._.ww...~.......~...n.wn_.»_m.v.......- was?Amm...mammalMuwWw.mmuww_mmvuM__~W-..mimgm _,WWWM.WM_._~W.
H \ 4 ' V Z a? - k Q E
w . TI. . -- 2 % K: .
Naming ighmikfuiwmgzw if
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 make use of the following theorem about symmetric polynomi