Algebra I MATH235
2
C ONTENTS
Part 1. Some Language and Notation of Mathematics
1. Sets
1.1. First definitions
1.2. Algebra of set operations
2. Proofs: idea and technique
2.1. Proving equality by two inequalities
2.2. Proof by contradiction and the contr
Math 235 Assignment 7 Solutions
November 6, 2015
4
a) x2 2 is irreducible in Q[x]. Indeed, since it is a quadratic polynomial, it either
has a root in Q or is irreducible. By Proposition 17.5.3, the only possible roots
are 1 or 2, none of which work. Ther
Math 235 Assignment 5 Solutions
October 20, 2015
1) Here are examples of relations of the following types:
reflexive, symmetric, transitive: Any equivalence relation satisfies these properties. For example, let A be the set of real numbers. We define a r
Math 235 Assignment 8 Solutions
November 16, 2015
1) Let R be a ring. We want to show that there exists a unique ring homomorphism
f : Z R.
Define
f (n) =
1R + 1 R + + 1 R ,
cfw_z

for n > 0
0R ,
f (n),
for n = 0
for n < 0
n times
It is easy to see that
SOLUTIONS FOR ASSIGNMENT SIX
JASON K.C. POLAK
A BSTRACT. Questions to be done: p. 50: 12, p. 61: 1,2,3, prove R[ x ] is a commutative ring if R is a
commutative ring.
C ONTENTS
1. p. 50: 12
2. p. 61: 1
3. p. 61: 2
4. p. 61: 3
5. R[ x ] is a Ring
1
1
2
3
4
SOLUTIONS FOR ASSIGNMENT NINE
JASON K.C. POLAK
C ONTENTS
1. p. 80, #5
2. p. 81, #9
3. P. 81, #15
3.1. Part (a)
3.2. Part (b)
3.3. Part (c)
3.4. Part (d)
4. P. 82, #19
5. Pairwise Nonisomorphic Rings
1
2
2
2
2
3
3
4
4
1. P. 80, #5
(a)
(b)
(c)
(d)
(e)
Prove
December 2012
Final Examination
McGill University
Faculty of Science
Abstract Algebra
Math 235
December, 2012
Time: 9:00 am  12:00 pm
Examiner: Prof. Henri Darmon
Associate Examiner: Prof. Eyal Goren
INSTRUCTIONS
1. Please answer questions in the exam bo
Basic Algebra 1
Solutions to Practice Final
Bahare Mirza
December 10, 2012
(1) We want to compute the reduced residue of 3N +1 where N = some big
power of 10. Note that the following works as long as the power of 10 is bigger
than 0, and not just in the p
SOLUTIONS FOR ASSIGNMENT THREE
JASON K.C. POLAK
C ONTENTS
1. Question One (Notes, 11.1)
2. Question Two (Notes, 11.4)
3. Question Three (Notes, 11.5)
4. Question Four (Notes, 11.6)
5. Question Five (Notes, 11.7)
6. Question Six (Supplemental Question One)
Algebra I MATH235
2
C ONTENTS
Part 1. Some Language and Notation of Mathematics
1. Sets
1.1. First definitions
1.2. Algebra of set operations
2. Proofs: idea and technique
2.1. Proving equality by two inequalities
2.2. Proof by contradiction and the contr
Assignment 11, MATH 235, Fall 2015
Eyal Goren
December 7, 2015
Do not submit this assignment.
Answer the following questions from pages 106107: (11), (12), (13), (14).
In addition, solve the following questions.
(a) Suppose that S3 acts on a nonempty se
Examiner: Professor E. Goren
Associate Examiner: Professor H. Darmon
91:59
6.
McGILL UNIVERSITY
FACULTY OF SCIENCE
FINAL EXAMINATION
MATH 235
ALGEBRA 1
Date: Friday December 22, 2006.
Time: 9:00 AM 12:00 PM
INSTRUCTIONS
. Please answer all ques
Chapter 2: Arithmetic in Z
Division with residue
Definition: a divides in Z
Properties
Corollary: b divides a iff a=bq+r where r=0
GCD
Definition
Theorem 9.1.2
Corollary
Euclidean Algorithm (practical)
Theorem and proof
Primes and Unique Factorization
Def
Math 235 Assignment 4 Solutions
October 7, 2015
8
a) Suppose a  k and b  k. Using the division algorithm, write k = q[a, b] + r
where 0 r < [a, b]. Since a  k and a  [a, b] (by definition), it follows that
a  r. In the same way we find that b  r. Si
Chapter 2: Arithmetic in Z
Division with residue
Definition: a divides in Z
Properties
Corollary: b divides a iff a=bq+r where r=0
GCD
Definition
Theorem 9.1.2
Corollary
Euclidean Algorithm (practical)
Theorem
Math 235 Assignment 10 Solutions
December 8, 2015
2) Are the following sets subgroups?
(a) H = cfw_1, (12)(34), (13)(24), (14)(23). This is a subgroup:
i. It contains the identity.
ii. We want to show that H is closed under multiplication. First we verify
Name :_ Student number:
VERSION A
MCGILL UNIVERSITY
FACULTY OF SCIENCE
FINAL EXAMINATION
MATH 235
ALGEBRA 1
Examiner: Professor E. Goren _ Date: Wednesday December 9, 2009.
Associate Examiner: Professor J. Loveys Time: 2:00 PM 5:00 PM
A
one!
INSTRUCTIONS