Problem 4 that 1 + 2 + 3 is the class equation of D3 which is isomorphic
to S3 , thus S3 indeed has 3 conjugacy classes.
Answer :cfw_1, I2 , I3 and S3 .
Part II. Rings and elds
7. Let F = cfw_a + b 19 : a, b Q C.
(a) Show that R is a ring, R F and F is a
Part II. Rings and elds
The following set of problems is to show that the ring
1+ 19
,
2
R = Z[] = cfw_a + b : a, b Z,
where =
is a principal ideal domain (PID) that is not a Euclidean
domain (ED) (a result of Motzkin).
7. Let F = cfw_a + b 19 : a, b Q C.
Math 4320 : Introduction to Algebra
Prelim II (Chapter 2& 3)
(due April 17 , 2009, 1:25pm)
You are not allowed to discuss your answers with others. Books, lecture notes and
calculators are allowed.
Part I. Group theory
1. (2.88) Show that a nite group G g
Math 4320 : Introduction to Algebra
First Prelim
(February 27, 2009, 1:25pm-2:15pm)
1. (5 pts each) Dene each of the following groups by explicitly describing
the set and the operation, and answer the questions without proof.
(i) Dene the symmetric group
Math 4320 : Introduction to Algebra
Prelim II (Chapter 2& 3)
(due April 17 , 2009, 1:25pm)
You are not allowed to discuss your answers with others. Books, lecture notes and
calculators are allowed.
Part I. Group theory
1. (2.88) Show that a nite group G g
Part II. Rings and elds
The following set of problems is to show that the ring
1+ 19
,
2
R = Z[] = cfw_a + b : a, b Z,
where =
is a principal ideal domain (PID) that is not a Euclidean
domain (ED) (a result of Motzkin).
7. Let F = cfw_a + b 19 : a, b Q C.
Teddy Einstein
Math 4320
HW2 Solutions
Problem 1: 2.22
Find the sign and inverse of the permutation shown in the book (and below).
Proof. Its disjoint cycle decomposition is:
(19)(28)(37)(46)
which immediately makes it an even permutation because it is a
Math 4320 Introduction to Algebra. Homework 12
An additional question:
1. Show that Z[ 5] = a + b 5 | a, b Z is a domain but not a UFD. (Hint: for the
second part, nd two dierent factorizations of 6, and remember to show that the factors
you give are irre
Math 4320 Introduction to Algebra. Homework 11
Extra questions.
1. Express x8 1 and x10 1 as products of cyclotomic polynomials i (x). Express each
i (x) you encounter as a polynomial in x.
2. Find all the elements which generate the multiplicative group
5
there are 5 (4 + 1) = 35 choices for products of two transpositions and single transpositions which
2
move 1. This makes a total of 50 such permutations which move 1.
Now count permutations xing 1 but moving 2. These must be products of 1 or 2 transposi
4
which contradicts the inductive hypothesis. On the other hand, if (a, b) is not an inversion of ,
then 1 has n + 1 inversions. Since the preceding lemma shows that multiplying on the right by
a simple transposition changes the number of inversions by at
3
Proof. First we claim if (j, k ) is an inversion of , with j < k and j, k i, i + 1, then (j, k ) is an
inversion of . Since xes j, k , we see that (k ) = (k ) and (j ) = (j ), so (j ) = (j ) >
(k ) = (k ) since (j, k ) is an inversion of . Hence (j, k
2
Suppose on the other hand that is a power of an n-cycle. Write = (r1 , . . . , rn )m . Without
loss of generality, we may assume that 0 m < n because n = (1). Dene r for all Z so that for
1 i n, r = ri if and only if r ri (mod n). Observe that k (r ) =
Math 4320 Prelim 2 Solutions
The main ideas plus common mistakes (but not all the details) are listed
below.
1. Let K = F5 [x]/(x3 + x2 + 2).
(a) Prove that K is a eld with 125 elements.
Solution: Check by plugging in all possible values that x3 + x2 +2 h
Math 4320 Prelim, Part I
1:25pm2:15pm, Monday 12th March 2012
Algebra is generous; she often gives more than is asked of her. Jean-Baptiste le Rond DAlembert
This exam contains three questions. Choose ONLY TWO to answer if you attempt
more than two questi
Math 4320 : Introduction to Algebra
Final Exam
(May 15, 2009, 2-4pm)
Books and calculators are not allowed.
1. (basic notions in group theory)
(a) (5 pts) Is D2n an abelian (commutative) group? How about Sn ? (Your
answer might depend on n.)
D2n and Sn ar