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 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 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 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 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
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 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 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 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 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 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
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 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.
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 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 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 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
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
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
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
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
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
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
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
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
MATH 228 Introduction to ring theory:
Part 6
Solving equations in Zn
As we have seen many properties of the arithmetic of integers (such
as commutative law, the identity elements and so on) carry over to the
modular case as well. However one needs to be c
Introduction to Ring Theory 228: Assignment 5
(Due in class Wednesday June 8)
1. (5 points) Let R be a ring. What are the units of R[x]? Justify.
2. (5 points) Let f (x) = an xn + an1 xn1 + + a0 Q[x], such f (x) has
all coecients in Z. Suppose that r/s Q
MATH 228 Introduction to ring theory:
Part 6
Rings
We have seen that many rules (such as associative law, distributive
law, commutative law) of ordinary arithmetic hold not only in Z but
also in Zn . You know other mathematical systems, such as the real
n
MATH 228 Introduction to ring theory:
Part 5
Corollary. Let A and B be two congruence classes. Then A and B
either equal or they are disjoint (i.e. dont have common elements).
Proof. Since A and B are congruences classes we have A = [a] and
B = [b] for so
MATH 228 Introduction to ring theory:
Part 4
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 r < b is the remainder. Then
g