MTH 310-1
Abstract Algebra I and Number Theory
Syllabus
S15
1/12/15
Instructor: Ulrich Meierfrankenfeld
Oce: C206 Wells Hall
E-mail: meier@math.msu.edu
Web page: www.math.msu.edu/meier
Oce Hour: Mo 1:50-2:40, Wed 1:50-2:40, Th 1:00-3:00, Fr 10:20-11:10 +
MTH 310-3
Abstract Algebra I and Number Theory
S15
Homework 2/ Solutions
Exercise 1. Prove that the given function is injective
(a) f : Z Z, x 2x.
(b) f : R R, x x3 .
(c) f : Z Q, x x .
7
(d) f : R R, x 3x + 5.
According to Theorem 1.3.13(b) we need to sh
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 1/Solutions
Exercise 2 Use a truth table to verify the statements (LR 17), (LR 26), (LR 27) and (LR 28) from the
rst exercise.
LR 17:
not -(P = Q) (P and not -Q).
P
T
T
F
F
Q P = Q not -(P = Q)
T
Final Exam: Thursday, May 7, 12:45-2:45pm in A201WH
MTH 310-1
Abstract Algebra I and Number Theory
S15
Review for Final Exam
Part I
The exam will be Open Book and Open Online Lecture Notes. Nevertheless, I recommend that
you memorize the following:
Deniti
MTH 310-1
Abstract Algebra I and Number Theory
S15
Review for Exam 2
Part I
The exam will be Open Book and Open Online Lecture Notes (this does NOT include the solutions
of the homework problems). Nevertheless, I recommend that you memorize the following
MTH 310-1
Abstract Algebra I and Number Theory
S15
Review for Exam 1
Part I
The exam will be Open Book and Open Online Lecture Notes (this does not include solutions to the
homework problems. Nevertheless, I recommend that you memorize the following theor
Final Exam: Thursday, May 7, 12:45-2:45pm in A201WH
MTH 310-1
Abstract Algebra I and Number Theory
S15
Practice Final Exam
Justify all your answers
1. Compute the remainder of 26004 when divided by 13.
2. Let R be a ring and T R. Put
S = cfw_a R | at = ta
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 8
due on 4/3/15
Exercise 1 Let R be a ring with identity and a, b, c R. Suppose that a is a unit in R.
(a) Show that
ab
b
ba
= ac
= c
= ca
ab =
b =
ba =
(b) Show that
0R
0R
0R
Exercise 2 Let R be
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 11
due on 4/24/15
Exercise 1 Let p = x3 + x2 + 1 Z2 [x].
(a) Determine the addition and multiplication table of Z2,p [].
(b) Is Z2,p [] a eld?
(c) Find all roots of p in Z2,p [].
Exercise 2 Let p
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 10
due on 4/17/15
Exercise 1 Let F be a eld and f F [x]. Show that
(a) If deg f = 1, then f is irreducible and f has a root in F .
(b) If deg f 2 and f is irreducible, then f has no root in F .
(
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 9
due on 4/10/15
Exercise 1 Let F be a eld and f, g, h F [x] with 1F = gcd(f, g). If f | h and g | h, prove that f g | h.
Exercise 2 Let F be a eld and f, g, h F [x]. Suppose that g = 0F and 1F =
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 7
due on 3/20/15
a
b
a, b Z2 . Given that S is a subring of M2 (Z2 ). Show that S is iso0 a+b
morphic to the ring R in Exercise 1 of Homework 3.
Exercise 1 Let S =
Exercise 2 Let f : R S and g :
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 5
due on 2/27/15
Exercise 1 Let R be a ring and I a set. Let Fun(I, R) be the set of functions from I to R. For f, g
Fun(I, R) let f + g and f g be the functions from I to R dened by
(f + g)(i)
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 4
due on 2/13/15
Exercise 1 . Prove or give a counterexample:
Let R be a ring and a, b R. Then
(a + b)2 = a2 + 2ab + b2 .
(Note here that according to Denition 2.3.3(c) 2d = d + d for any d in R.
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 6
due on 3/6/15
Exercise 1 Let a, b and e be integers. Suppose a and b are not both zero and that e is a positive common
divisor of a and b. Put d = gcd(a, b).
(a) Prove that gcd
a b
e, e
= d.
e
MTH 310-3
Abstract Algebra I and Number Theory
S15
Homework 2
due on 1/30/15
Exercise 1. Prove that the given function is injective
(a) f : Z Z, x 2x.
(b) f : R R, x x3 .
(c) f : Z Q, x x .
7
(d) f : R R, x 3x + 5.
Exercise 2. (a) Let f : B C and g : C D
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 4/ Solutions
Exercise 1 Prove or give a counterexample:
Let R be a ring and a, b R. Then
(a + b)2 = a2 + 2ab + b2 .
(Note here that according to Denition 2.3.3(c) 2d = d + d for any d in R.
Solut
MTH 310-3
Abstract Algebra I and Number Theory
S15
Homework 2/ Solutions
Exercise 1. Let R = cfw_0, e, b, c with addition and multiplication dened by the following tables. Assume
associativity and distributivity and show that R is a ring with identity. Is
Final Exam: Thursday, May 7, 12:45-2:45pm in A201WH
MTH 310-1
Abstract Algebra I and Number Theory
S15
Review for Final Exam/ Solutions
1. Is 12400 10288 divisible by 11?
We compute in Z11 :
12 = 12 11 = 1 and 10 = 10 11 = 1. So 12400 = 1400 = 1 and 10288
MTH 310-1
Abstract Algebra I and Number Theory
S15
Review for Exam 2/Solutions
1.
Let n be an integer.
(a) For 0 n 10, verify that n2 n + 11 is a prime.
(b) If n = 0, 1 and n 0, 1 (mod 11), show that n2 n + 11 is not a prime.
(c) Prove or disprove: If n 0
Final Exam: Thursday, May 7, 12:45-2:45pm in A201WH
MTH 310-1
Abstract Algebra I and Number Theory
S15
Practice Final Exam / Solutions
1. Compute the remainder of 26004 when divided by 13.
We compute in Z13 :
22 = 4,
23 = 8,
24 = 16 = 3,
25 = 2 23 = 2 3 =
MTH 310-1
Abstract Algebra I and Number Theory
2/18/2015
Practice Exam I/Solutions
1.
Let be the relation on Q dened by a b if a + b Z. Show that is not transitive.
Note that
1 3
4
3 5
8
+ = = 1 Z,
+ = =2Z
4 4
4
4 4
4
Thus the denition of shows that
1
3
MTH 310-1
Abstract Algebra I and Number Theory
S15
Practice Exam 2/Solutions
1.
Let n be an odd integer. Show that gcd(n + 1, n 1) = 2.
By Theorem 2.9.4 d = gcd(a, b) whenever a, b, q, r.d are integers, d = gcd(q, r) and b = aq + r. As
n + 1 = (n 1) 1 + 2
Your Name:
MTH 310-1
Abstract Algebra I and Number Theory
S15
Exam 2/Solutions
1.
Let a and b be integers and let e be a positive integer. Show that e = gcd(a, b) if and only if
(1) e is a common divisor of a and b in Z, and
(2) there exist u, v Z with e
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 11/Solutions
Theorem A Let R be a commutative ring with identity and suppose that 1R + 1R = 0R . Let a, b, c R.
Then
(a) a + a = 0R .
(b) a = a.
(c) (a + b)2 = a2 + b2 .
(d) (a + b + c)2 = a2 + b
MTH 310-1
Abstract Algebra I and Number Theory
Exam I
Solutions
2/20/2015
1. Let A and D be sets and f : A D a function. Suppose is an equivalence relation on D and let
be the relation on A dened by
ab
f (a) f (b).
for all a, b A. Show that is an equival
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 10/Solutions
Exercise 1 Let F be a eld and f F [x]. Show that
(a) If deg f = 1, then f is irreducible and f has a root in F .
(b) If deg f 2 and f is irreducible, then f has no root in F .
(c) If
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 8/Solutions
Exercise 1 Let R be a ring with identity and a, b, c R. Suppose that a is a unit in R.
(a) Show that
ab
b
ba
= ac
= c
= ca
ab =
b =
ba =
(b) Show that
0R
0R
0R
(a) Since a is a unit,
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 9/Solutions
Exercise 1 Let F be a eld and f, g, h F [x] with 1F = gcd(f, g). If f | h and g | h, prove that f g | h.
I will give two proofs:
Proof 1: Since 1F = gcd(f, g) we conclude from 3.2.14