PMath 346 Solutions to Assignment 6 (Corrected)
1. WLOG, By the proof of Problem 3, G is a semidirect product
G C7
=
C3
By Lagrange, the only possible orders of elements are cfw_1, 3, 7, 21. There is a unique
element of order 1. No element of G can have o

PMath 346 Solutions to Assignment 5
1. Each Sylow p-subgroup is of order p, and is thus isomorphic to Cp since p is prime. Each
such Sylow p-subgroup has (p 1) elements of order p. Sp has p! = (p 1)! elements of
p
order p. Any two distinct Sylow p-subgrou

PMath 346 Solutions to Assignment 4
1. Theorem 1 Let G be a group and H G a subgroup with [G : H ] = m, for some positive
integer m. Then there exists a normal subgroup N G, N H with m [G : N ] m!.
Proof: Let G act on the set of left cosets of H in G by l

PMath 346 Solutions to Assignment 3 (Corrected)
Before we solve the assignment problems, we prove a lemma.
Lemma 1 Let G be a group and N, H G. Then
1. If N G, then HN is a subgroup of G.
2. If N, H G, then HN G.
3. N H is a subgroup of G.
4. If N G, then

PMath 346 Solutions to Assignment 2
1. (a) Suppose for a contradiction that Q2 is cyclic. Then there exists a generator
g = 2a Q2 , where g = Q2 . So since 2n1 Q2 , there must exist an integer k such
n
+1
that
1
= kg
2n+1
a
= k( n )
2
1 = 2ka
clear denomi

PMath 346 Solutions to Assignment 1
1. G is not a group with the given operation. The associative axiom is not satised, for
example,
2 (3 2) =
=
(2 3) 2 =
=
2 (5)
3
(1) 2
1
2. (a) We have the following table for D14 :
order
number of elements
1 2 7 14
1 1

PMATH 346, Winter 2011
Assignment #6
Due at 8:30am on Friday, April 1st, 2011
Problem 1. Let G be a non-abelian group of order 21. State the number of elements in G of each
possible order (i.e., how many elements of order 1, how many elements of order 2,

PMATH 346, Winter 2011
Assignment #5
Due at 8:30am on Friday, March 18th, 2011
Problem 1. If p is an odd prime, how may Sylow p-subgroups are there in Sp ?
Problem 2. Describe, up to isomorphism, all abelian groups of order 144.
Problem 3. Let G = C3 C3 C

PMATH 346, Winter 2011
Assignment #4
Due at 8:30am on Friday, March 4th, 2011
Problem 1. Use the class equation to prove that a group of order 726 cannot be simple 2 3 112 .
Problem 2. Which is the smallest n such that D5 is isomorphic to a subgroup of Sn

PMATH 346, Winter 2010
Assignment #3
Due at 8:30am on Friday, February 11th, 2011
Problem 1. The exponent of a nite group G is the smallest positive integer t such that g t = 1 for all
g G. Prove that the exponent of Sn is lcm(1, 2, ., n).
1
Problem 2. Le

PMATH 346, Winter 2010
Assignment #2
Due at 8:30am on Friday, January 28th, 2011
Problem 1. Let Q2 = cfw_a/b Q : a, b Z, b a power of 2.
(a) Prove that Q2 is not cyclic.
(b) Prove that if H is a proper subgroup of Q2 which contains Z, then H is cyclic.
Pr

PMATH 346, Winter 2010
Assignment #1
Due at 8:30am on Friday, January 14th, 2011
Problem 1. Let G = cfw_1, 2, 3, 4, 5 be a set with
1
11
22
33
44
55
Is G a group? Justify your answer.
a binary
234
234
415
542
153
321
operation dened as the following:
5
5

PMath 345 Spring 2010
Polynomials, Rings, and Finite Fields
Detailed Course Outline
The third column represents page numbers in the textbook(s) corresponding
to the topic to be covered that day in lecture, described in the fourth column.
Page ranges marke

PMath 345 Homework 8 Solutions
1. Let R be an integral domain with the property that every integral ideal is
invertible. (Recall that an integral ideal is an actual ideal of R, as opposed
to a fractional ideal of R.) Prove that every fractional ideal of R

PMath 345 Homework 7 Solutions
1. Find all the roots of x2 + x 1 in the eld GF(9) (Z/3Z)(i).
=
Solution: Well use the quadratic formula:
1 1 + 4
x=
=1i
2
So the roots are 1 + i and 1 i. That was easy.
Alternatively, you could take a + bi and plug it into

PMath 345 Homework 6 Solutions
1. Compute the following degrees:
a) [Q( 6 2) : Q( 2)]
Solution: The degree is 3.
To see this, note that p(x) = x3 2 is a polynomial with coecients in
Q( 2) which has 6 2 as a root, so the degree is at most 3. If we check to

PMath 345 Homework 5 Solutions
1. Consider the polynomial ring R[x.y ] with the lexicographic ordering induced by x > y , as described in class. Assume that the set cfw_x3 y, x2 y
y 2 , xy 2 y 2 , y 3 y 2 is a Grbner basis of the ideal I = (x3 y, x2 y y

PMath 345 Homework 4 Solutions
1. Which of the following polynomials are irreducible? Prove your answers.
(a) x3 + 4x + 7 in Z[x].
Solution: Irreducible. If you reduce the polynomial modulo 5, it becomes
x3 x + 2. A quick check shows that it has no roots

PMath 334 Homework 3 Solutions
1. Let D = Z[1/2] = cfw_a/2n | a, n Z, the ring of dyadic rationals. Prove
that the fraction eld of D is Q.
Solution: Your rst instinct might be to try to do this constructively, and
indeed it can be done that way. But there

PMath 345 Homework 2 Solutions
1. Prove that the ideal (x) in Q[x] is maximal.
Solution: Say that I is an ideal of Q[x] which contains (x), but is strictly
bigger. Then there is some polynomial p(x) I with p(x) (x). That
means that p(x) has a nonzero cons

PMath 345 Homework 1 Solutions
1. The set cfw_0, 2, 4 under addition and multiplication modulo 6 has a unity.
Find it.
Solution: The unity is 4. The reason for this is simply that 4 0 = 0,
4 2 = 2, and 4 4 = 4, all modulo 6. Since multiplication by 4 neve