Solutions to Exercises of MATH3131 (T05)
CHAN, Chin Hei, Terry
9th March, 2015
14.
Proof. First note that if a is an element of order r in G, then the subset H = cfw_1G , a, a2 , ., ar1 is a subgroup of order
r in G, and by the Lagranges Theorem, we see

Algebra Exercises 3
K F Lai
March 19, 2015
3. Fields.
1. Draw pictures to illustrate and answer the following questions.
(1) Given cfw_0, 1 and positive real numbers cfw_a, b let 1 be he line passing
through (1, 0) and (0, a). Line the line through (b, 0)

The polynomial ring Z[x] (Exercise 3 of MATH3131)
CHAN, Chin Hei, Terry
17th April, 2015
Definition 1. Given a positive
integer n, we say two integers x and y are congruent modulo n,
written as x y(mod n), if n (x y).
One can easily check that (mod n) is

Algebra Exercises 1
K F Lai
February 6, 2015
Groups.
1. Let G be a group. Show that (1) if x G then xG = G, (2) G G = G.
2. Let H be a subgroup of G. From
xHyH = xyy 1 HyH
deduce that if H is a normal subgroup then xHyH = xyH.
3. Let : G H be a map such t

MATH3131 Honours Linear and Abstract Algebra II (Spring 2015)
Final Examination
CHAN, Chin Hei, Terry
May 20, 2015
Date: 30th May, 2015
Time: 12:30-14:30
Venue: Room 3007 (3rd floor near Lift 3)
Instructions:
1. Please do NOT come into the classroom if yo

Solutions to Exercises of MATH3131 (T06)
CHAN, Chin Hei, Terry
16th March, 2015
16.
(123)(12) = (13), (12)(123) = (23), hence they are not the same, or, in other words, (123) does not commute with
(12) in S3 .
Note that up to isomorphism, the only group o

Brief Solutions to Exercises of MATH3131
CHAN, Chin Hei, Terry
18th April, 2015
Fields.
5.
(2 + + 1)(2 + ) = 2 2
2
3
1
( 1)1 = 2
5
5
5
7.
First write a + b
2 + c 3 +
d 6 = 0, a, b, c, d
Q.
Then noting that 6 = 2 3, we have a + b 2 + 3(c + d 2) = 0.
2