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. Neverthele
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
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). Nev
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 accordin
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,
MTH 310-1
Abstract Algebra I and Number Theory
Syllabus
S15
1/12/15
Instructor: Ulrich Meierfrankenfeld
Oce: C206 Wells Hall
E-mail: [email protected]
Web page: www.math.msu.edu/meier
Oce Hour: Mo 1:
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 dis
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
a
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 -(
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 7/Solutions
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.
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
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 el
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 de
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 b
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 Homewor
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 an
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 ac
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.
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 :
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 3
due on 2/6/15
Exercise 1 Let R = cfw_0, e, b, c with addition and multiplication dened by the following tables. Assume
associativity and d
Your Name:
MTH 310-1
Abstract Algebra I and Number Theory
Exam 2
Justify all your answers
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
Your name:
MTH 310-1
Abstract Algebra I and Number Theory
Exam I
Justify all your answers
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 t
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 1
due on 1/23/15
Exercise 1 Let P , Q and R be statements, let T be a true statement and F a false statement. Convince
yourself that each of
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. Neverthe
MTH 310-1
Abstract Algebra I and Number Theory
S15
Homework 5/ Solutions
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
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 1
MTH 310-1
Abstract Algebra I and Number Theory
S15
Review for Exam 1/Solutions
1.
Find all solutions of the equation x3 = 1 in Z7 .
We compute in Z7
03
13
23
33
43
53
63
=
8
=
27 = 1
= (3)3 = 33
= (2)
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
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,
+