Math 412 - Introduction to Abstract Algebra
Homework 7 solutions
1. (5.1.8) Prove of disprove. If p(x) is relatively prime to k(x) and f (x)k(x)
g(x)k(x) mod p(x), then f (x) g(x) mod p(x).
By denition, f (x)k(x) g(x)k(x) mod p(x) means p|(f k gk). If
p|
Math 412 - Introduction to Abstract Algebra
Homework 9
This homework assignment concerns sections 7.3, 7.4, and 8.1 in the text.
Please turn the following seven problems in on Wednesday, April 16.
1. (7.3.42) Let k be a positive divisor of the integer n.
Math 412 - Introduction to Abstract Algebra
Homework 8 solutions
This homework assignment concerns sections 7.1-3 in the text. Please turn
the following seven problems in on Friday, April 4.
1. (7.1.28) Prove that each element of a nite group appears exac
Math 412 - Introduction to Abstract Algebra
Homework 8
This homework assignment concerns sections 7.1-3 in the text. Please turn
the following seven problems in on Friday, April 4.
1. (7.1.28) Prove that each element of a nite group appears exactly once i
Math 412 - Introduction to Abstract Algebra
Homework 9 solutions
This homework assignment concerns sections 7.3, 7.4, and 8.1 in the text.
Please turn the following seven problems in on Wednesday, April 16.
1. (7.3.42) Let k be a positive divisor of the i
Math 412 - Introduction to Abstract Algebra
Final Exam - Extra problems solutions
Here are partial solutions to the list of problems I gave on the last day of
class to prepare for the nal exam.
1. (8.2.18) If K, N are normal in G, prove that K N is normal
Math 412 - Introduction to Abstract Algebra
Homework 7
This homework assignment concerns sections 5.1-3 in the text. Please turn
the following seven problems in on Wednesday, March 19.
1. (5.1.8) Prove of disprove. If p(x) is relatively prime to k(x) and
Math 412 - Introduction to Abstract Algebra
Homework 6 solutions
1. (4.5.12) To prove the contrapositive, we assume that f (x) is reducible.
This implies that f (x) = g(x)h(x) for all x F . In particular, for the
translation x x + c, we have f (x + c) = g
Math 412 - Introduction to Abstract Algebra
Homework 5 solutions
1. (4.3.15) If a and b are distinct, then there are p such polynomials (since
2
order does not matter). If a = b, then there are p such polynomials. We
have p + p = p(p 1)/2 + p = (p2 + p)/2
Math 412 - Intorduction to Abstract Algebra
Homework 4 solutions
1. (4.1.13) If f (x) is a zero divisor, then there exists a non zero polynomial
g(x) = b0 + b1 x + + bm xm (with bm = 0R ) such that f (x)g(x) = 0R[x] .
The product expansion of f (x)g(x) ha
Math 412 - Introduction to Abstract Algebra
Homework 6
This homework assignment concerns sections 4.5 and 4.6 in the text. Please
turn the following seven problems in on Wednesday, March 12.
1. (4.5.12)
2. (4.5.15)
3. (4.5.20)
4. (4.5.21)
5. (4.6.5)
6. (4
Math 412 - Introduction to Abstract Algebra
Homework 5
This homework assignment concerns sections 4.3 and 4.4 in the text. Please
turn the following seven problems in on Wednesday, March 12.
1. (4.3.15)
2. (4.3.16)
3. (4.3.20)
4. (4.4.14)
5. (4.4.19)
6. (
Partial Solutions for Homework #3
Math 412, Winter 2014
3.1.18) Claim: If we dene a new multiplication on Z by setting ab = 1 for
all ab Z, then Z with this new multiplication (and ordinary addition) is
not a ring.
Proof: One may check that the distributi
More Examples on Proof Writing
Here are two more examples of simple proofwriting exercises. We will approach them in the
manner of the Tips on Proof Writing handout.
Example 1. Prove: Let a, b Z, and let m > 0 be an integer. Then
gcd(ma, mb) = m gcd(a, b)
Transformadas integrales
Prof. Enrique Zamora Gallardo
Facultad de Ingeniera, Universidad Anhuac Mxico Norte
Tarea N 2, Fecha de entrega: Mircoles 31 de agosto 13:00
Nmeros Complejos
1. Suponga que el producto de dos nmeros complejos es cero. Demuestre qu
MAT 412 HOMEWORK 5
SOLUTIONS
DUE: FEBRUARY 10, 2017 (BEGINNING OF CLASS)
This homework set covers sections 3.2, 3.3. References are to Hungerford, 3rd. edition.
Problem 1. (a) Show that R = Z[ 5] = cfw_ a + b 5 : a, b Z is a subring of R. Is R a
field?
(b
MAT 412 HOMEWORK 7
SOLUTIONS
DUE: MARCH 10, 2017 (BEGINNING OF CLASS)
This homework set covers sections 4.6, 5.1 and 5.2. References are to Hungerford, 3rd. edition.
Problem 1. (a) Factor the polynomials x2 + x 2 and x3 x2 + 2x 2 in R[ x ].
(b) Find two p
MAT 412 HOMEWORK 8
SOLUTIONS
DUE: MARCH 17, 2017 (BEGINNING OF CLASS)
This homework set covers sections 5.3, 6.1. References are to Hungerford, 3rd. edition.
Problem 1. (a) Show that Z2 [ x ]/( x3 + x + 1) is a field that contains all 3 roots of x3 + x +
Name; ELEOUOQE 4346R
MAT 412 QUIZ 1
JANUARY 11, 2016
Problem 1. (1 point) Consider the relation
r = cfw_(a,b) e z x z : (1) = (1)".
Is T an equivalence relation on 2? (No explanation necessary! BUI': get a bonus point for the
' ' tification of your answer
Name: @LLW W
MAT 412 QUIZ 2
JANUARY 18, 2016
Problem 1. (3 points) TRUE / FALSE (no explanation neededl):
Let a, b, u, v be integers.
(a) If au + bv = d then (a,b) = d.
ALS (6%: ,s.gd+gg.c.9v)=ghw (510:4)
& 0L 6' d,
ll
U'
(b) If a b = 11u, then a and b le
WW
MAT 412 QUIZ 3
JANUARY 25, 2017
Problem 1. (3 points) TRUE/ FALSE (no explanation needed!):
(a) Let n > 1 be an integer. Then here are exaclty n - 1 congruence classes modulo n.
[OJ/"'ID-G]
jr'ALSE " [Wm an M (MWCQ clam modn:
(b) Let n > 1 be an intege
Chapter 1
Homework typesetter
The five .tex files Typeset, Instructions, Homework1,2,All which I wrote and
added to FILES on CANVAS comprise a homework typesetter called Typeset.tex and the other 4 files, files which it typesets (compiles), upon your comm
\sectioncfw_Instructions for using Typeset.tex
\begincfw_enumerate
\item Lines in Typeset.tex which contain \emphcfw_input are \emphcfw_typeset lines,
lines calling for their inputs to be compiled.\
Those which begin with \% are closed, invisible to the c
SULIT
UNIVERSITI TEKNOLOGI MARA
FAKULTI KEJURUTERAAN
ELEKTRIK
Kod Kursus
Nama Kursus
:
:
SKIMA JAWAPAN PEMERIKSA
PEPERIKSAAN : JAN / TAHUN 2014
Pemeriksa
EPO 630
Pertama :
HIGH VOLTAGE
Pemeriksa
Kedua
:
No Soalan :
ANSWERS
Q1(a)
M
combine equation 1 and 2
1. Define the term of harmonics.
Harmonics are a mathematical way of describing distortion to a voltage or current waveform. The term
harmonic refers to a component of a waveform that occurs at an integer multiple of the fundamental
frequency. Fourier the
Partial Solutions for Homework #2
Math 412, Winter 2014
C.4) Let r be a real number such that r = 1. Prove that for every integer
n 1,
rn 1
1 + r + + rn1 =
.
r1
Proof: For all integers n 1, let P (n) be the mathematical statement
1 + r + + rn1 =
rn 1
.
r1