Introduction to Ring Theory 228, Assignment 4: Solutions
Problem 1: Using the Euclidean algorithm write the greatest common
divisor of a and b in the form ua + vb if
(1) a = 24, b = 138;
(2) a = 314, b = 159;
Solution. (1) We have
138 = 24 5 + 18
24 = 18
Introduction to Ring Theory 228: Assignment 1
Problem 1: Let f, g, h : Z Z be mappings defined by
f (n) = 2n, g(n) = n + 1, h(n) = n2 .
Find f f, h f, f g, g g, g h, h h.
Solution. We have
(f f ) = f (f (n) = f (2n) = 2(2n) = 4n;
(h f )(n) = h(f (n) = h(2
Introduction to Ring Theory 228: Assignment 4
For practice only
Problem 1 (5 points): Using the Euclidean algorithm write the greatest common divisor of a and b in the form ua + vb if
(1) a = 24, b = 138;
(2) a = 314, b = 159;
Problem 2 (5 points): Find t
Introduction to Ring Theory 228, Assignment 5: solutions
Problem 1: Express 5040 as a product of primes.
Solution. We have
5040 = 10 504 = (2 5) (4 126) = 23 5 (2 3 21) = 24 32 5 7.
Problem 2: Let a, b Z. Prove that g.c.d.(a, b) = 1 if and only if there i
MATH 228 Fall 2014
Homework 10
Due in class on Wednesday, November 26, 2014
NOTE: You will not be allowed to use your calculator during the midterm and the nal
exams. Therefore, you should not use it to do the homework.
Explain your answers as much as you
Introduction to Ring Theory 228: Assignment 5
Due in class, February 12
Problem 1: Express 5040 as a product of primes.
Problem 2: Let a, b Z. Prove that g.c.d.(a, b) = 1 if and only if
there is no prime p such that p | a and p | b.
Problem 3: A real numb
Introduction to Ring Theory 228: Assignment 3
Due in class, January 29, 2016
Problem 1 (5 points): Find the quotient and the remainder when you
divide a by b if:
(1) a = 302, b = 20;
(2) a = 2002, b = 17;
(3) a = 2002, b = 11.
Problem 2 (5 points): Applyi
Introduction to Ring Theory 228, Assignment 2: Solutions
Problem 1: Using the method of induction prove that for each positive
integer n > 0 one has 2n > n.
Solution. We first check that the statement is true for n = 1. Indeed, if
n = 1 then
2n = 21 = 2 >
Introduction to Ring Theory 228: Assignment 2
Due in class, January 22
Problem 1: Using the method of induction prove that for each positive
integer n > 0 one has 2n > n.
Problem 2: Let r be a real number, r 6= 1. Using the method of induction
prove that
Introduction to Ring Theory 228, Assignment 3: solutions
Problem 1: Find the quotient and the remainder when you divide a by b
if:
(1) a = 302, b = 20;
(2) a = 2002, b = 17;
(3) a = 2002, b = 11.
Solution. (1) We have
302 = 20 (16) + 18,
hence q = 16, r =
Introduction to Ring Theory 228: Assignment 1
Due in class, January 15
Problem 1: Let f, g, h : Z Z be mappings defined by
f (n) = 2n, g(n) = n + 1, h(n) = n2 .
Find f f, h f, f g, g g, g h, h h.
Problem 2: In each case find a formula for the inverse of a
I
1. (a) Let R be a commutative ring with multiplicative identity 1
=1=
O. Define units and
1II:
zero divisors of R.
(b) Find all units in R
= Z/(18).
(c) Compute 15-1 in Z7564'
2. Let f(x) = 5x4 + 3x3+ I and g(x) = 3x2 + 2x + I in Z7[X].Find the greatest
Math 228 Solutions 2
1. Since d is a common divisor of a and b, a/d and b/d are both integers. If c is a common
positive divisor of a/d and b/d, then dc divides a and b, hence dc|(a, b) by Corollary 1.4.
Since (a, b) = d, we have dc|d, hence c|1 and c =1.
Math 228 Solutions 4
1.Suppose ax + by = c for some integers x, y. Since (a, b) divides a and b, then it also
divides ax + by, hence also c.
Suppose that (a, b)|c, allowing us to write a = (a, b), (a, b), c = (a, b) for integers
, , . Then ax + by = c be
Math 228 Solutions 1
1.a) q = 117, r = 11 by the usual long division.
b) q = 16, r = 2. Using long division get 302 = 1915 + 17 and multiply by 1 to get
302 = 19( 15) + ( 17). This is close but not quite right because 0 17 < 19 isnt
true. But 17 = 19 + 2
Math 228 Solutions 3
1. Observe that a2 0 (mod 3) when 3|a, and that a2 1 (mod 3) otherwise: this is clear
when 3|a, and otherwise we have a = 3q + r with r = 1 or 2, hence a2 r2 1 (mod 3)
because 12 and 22 are both 1 (mod 3). The same property applies to
Math 228 Solutions 9
1. a) By trying all elements of Z5, x2 + 2 has no roots in Z5, so is irreducible in Z5[x].
b) x4 4 = (x2)2 22 = (x2 + 2)(x2 2) with x2 + 2 irreducible, by a), and x2 2 also
irreducible by the same process as in a).
2. For every p, the
Math 228 Solutions 7
1. Since f takes 1Z = 1 to 1Z, by Theorem 3.10.4, we have f(1) = 1. This starts an
induction that proves that f(n) = n, for all n 1: for now the induction hypothesis f(k) = k
implies f(k + 1) = f(k) + f(1) = k + 1, since f is a homomo
Math 228 Solutions 10
1. By the rational root test the only possible rational roots of x4 + 2x3 + x2 7x 18 are
the divisors of 18 in Z. Going through these we find that 2 is a root, hence x + 2 is a
factor. Long division produces the factorization (x + 2)
Math 228 Solutions 11
1. f(x + 1) = (x + 1)4 + 1 = x4 + 4x3 + 6x2 + 4x + 2 satisfies the Eisenstein criterion at the
prime 2, hence is irreducible in Q[x], by Theorem 4.24. Now Problem 10.6 with c = 1
implies that f(x) = x4 + 1 is irreducible in Q[x].
2.
Stat 266 Winter 2014 Assignment #4 Solutions Mike Kowalski
University of Alberta
Department of Mathematical and Statistical Sciences
Statistics 266, Winter 2014
Assignment #4
Assignment #4 (90 marks): Due Wednesday, April 9, at 10pm
1.
2.
3.
9.40
9.52
9.5
Stat 266 Winter 2014 Assignment #2 Solutions Mike Kowalski
University of Alberta
Department of Mathematical and Statistical Sciences
Statistics 266, Winter 2014
Assignment #2
Assignment #2 (90 marks): Due Thursday, February 27, at 10pm
1.
7.10
a.
(5 marks
MATH 228 Introduction to ring theory:
Part 11
Congruence classes in F [x]
Definition. Let F be a field and let p(x) F [x] be a nonzero polynomial. We say that two polynomials f (x) and g(x) are congruent
modulo p(x) if f (x) g(x) is divisible by p(x). Not
Introduction to Ring Theory 228: Assignment 9; Solutions
Problem 1: Let R be a ring. Which of the following subsets of R[x] are
subrings of R[x]? Justify your answers.
(a) All polynomial with constant term zero.
(b) All polynomials of degree 2.
(c) All po
Stat 266 Homework Winter 2014
Stat 266 Homework Assignments Winter 2014
- There are three homework assignments.
- Completed assignments should be submitted to a wooden box located on the third floor of CAB
labeled STAT 266 under your instructors name by 1
Math 228, Midterm Exam Practice
Question 1. (a) Use the Euclidean algorithm to find the greatest common
divisor of a and b when a = 23634 and b = 4338.
(b) Express this greatest common divisor as a linear combination of a and b.
In other words, find integ
MATH 228 Introduction to ring theory:
Part 4
Lecture 10, January 25, 2016.
We are almost ready to state Euclidean algorithm. We need only the
last observation.
Proposition. Let a, b be two nonzero integers and assume b is positive.
Let a = qb + r where 0
Introduction to Ring Theory 228, Assignment 6:
solutions
Problem 1: Let R be a set consisting of 4 elements: R = cfw_ 0, e, b, c .
Define addition and multiplication on R by the following tables:
+
0
e
b
c
0
0
e
b
c
e
e
0
c
b
b
b
c
0
e
0
e
b
c
c
c
b
e
0
0
Math 334 Homework #1
Due on Monday, January 23, 2017, by 5:00 pm, in the assignment box on the third floor of CAB.
I. Evaluate the following integrals
Z
cos t
1.
1 dt
(1 + sin t) 2
Z
dy
3.
2
y 4y + 8
Z
5.
sin2 x dx
Z
7.
e2x sin x dx
Z
9.
sin3 x cos4 x dx
Spring 2017
MATH 228
Practice Problems for Final Exam Solutions
1. (6 points)
(a) Let F be a field, and let f (x), g(x) and d(x) be non-zero polynomials over F .
What does it mean to say that d(x) is the greatest common divisor of f (x) and
g(x) in the ri
Spring 2017
MATH 228
Assignment 4 - Solutions
1. The set F = cfw_x + y 2 | x, y Q is a subring of the field of real numbers R (you may
take this as given). Show that F is, in fact, a field (Hint: You just need to show that
every non-zero element of F is a
Spring 2017
MATH 228
Assignment 3 - Solutions
1. Let a and b be (fixed, but unknown) integers. If
[4a] = [4b]
in Z2017 , does it follow that [a] = [b] in Z2017 ? Justify your answer.
Solution: The equation [4a] = [4b] (in Z2017 ) may be rewritten as
[4][a
Spring 2017
MATH 228
Practice Problems for Midterm Solutions
1. (6 points)
(a) Using the Euclidean algorithm, find the greatest common divisor of 217 and 154.
(b) Consider the set
S = cfw_x 217 + y 154 | x, y Z
of all integer linear combinations of 217 an
Spring 2017
MATH 228
Assignment 5 - Solutions
1. Let R be a ring, and let a, b R. Show that if b = ua for some unit u R, then
R a = R b (Note: R a and R b are the principal ideals of R generated by a and
b, respectively. To show that two sets are equal, s