MATH 310, MIDTERM EXAM SOLUTIONS
1. (a) State the denition of Zn .
Answer: The set Zn is obtained from Z by changing the denition of the
equals sign as follows:
a = b in Zn if and only if a = b + nq for some q Z.
Another way to express this idea is to dra
MATH 310, HOMEWORK 9
1. Recall that an n-pointed star is dened to be a diagram where we draw an arrow
from x to x + k for each x Zn . We know that there are exactly n dierent n-pointed
stars. Also, recall that a piece of a star is a part of the diagram th
MATH 310, HOMEWORK 8
1. Let a = (a0 , a1 , a2 , . . . ) be an innite sequence of real numbers. Consider the
operator E which multiplies the k -th term by k and then shifts the whole sequence one
step to the left:
E (a) = E (a0 , a1 , a2 , . . . ) = (a1 ,
MATH 310, HOMEWORK 7
1. (20 points) Suppose we distribute m objects among n bins in some way. If a bin
contains more than one object, we say it is overloaded. Write a careful proof for each
of the following statements:
(a) If m > n, prove that at least on
MATH 310, HOMEWORK 11
1. If S is a commutative ring with identity, let S 2 = cfw_(a, b) | a, b S with the following
addition and multiplication:
(a, b) + (c, d) = (a + c, b + d)
(a, b)(c, d) = (ac bd, ad + bc)
where a, b, c, d S . Recall that S 2 is a co
HOMEWORK 2 SOLUTIONS
MATT RATHBUN
MATH 310, SECTION 3
ABSTRACT ALGEBRA
1.3 #2, 6, 13, 23; 2.1 #2, 8, 11, 27
Section 1.3.
2. Suppose that p is prime, and let a be an integer. Then (a, p) is a positive divisor of p, so it is either
1 or |p|. If it is 1, the
HOMEWORK 9 SOLUTIONS
MATT RATHBUN
MATH 310, SECTION 3
ABSTRACT ALGEBRA
5.2 #2, 3, 5, 11; 5.3 #1, 2, 3
Section 5.2.
3. See solutions in the back of the text.
5. For any two equivalence classes of polynomials of the form [ax + b] and [cx + d], their product
HOMEWORK 10 SOLUTIONS
MATT RATHBUN
MATH 310, SECTION 3
ABSTRACT ALGEBRA
6.1 #1, 3, 4, 6, 8, 10, 11, 12, 13
Section 6.1.
1. See solutions in the back of the text.
3.
(a) First, we show that I is a subring of Z Z. First, observe that (0, 0) I , so I = . Sup
HOMEWORK 11 SOLUTIONS
MATT RATHBUN
MATH 310, SECTION 3
ABSTRACT ALGEBRA
6.1 #33, 38; 6.2 #4, 5, 8, 12, 13, 21, 23; 6.3 #3, 5, 11, 13
Section 6.1.
33. See solution in the back of the text.
38. Following the hint, suppose I is a non-zero ideal (since the ze
MATH 310, FINAL REVIEW
List of Denitions:
Note: You should understand these denitions thoroughly.
(1) Injective/surjective/bijective.
(2) Ring/ring with identity/commutative ring/commutative ring with identity.
(3) Invertible element of a ring with identi
HOMEWORK 8 SOLUTIONS
MATT RATHBUN
MATH 310, SECTION 3
ABSTRACT ALGEBRA
4.4 #7, 12, 13, 15, 17, 24; 5.1 #3, 5, 7, 12
Section 4.4.
7. Let f (x) = x7 x Z7 [x]. Then, note that f (0) = 0, f (1) = 0, f (2) = 0, f (3) = 0, f (4) = 0,
f (5) = 0, and f (6) = 0 mo
HOMEWORK 6 SOLUTIONS
MATT RATHBUN
MATH 310, SECTION 3
ABSTRACT ALGEBRA
4.1 #7, 11, 13, 16, 17, 18, 19; 4.2 #1, 2, 3, 7, 10, 14
Note: When we take two arbitrary polynomials in R[x], we cannot assume that they have the same
degree. However, the notation for
HOMEWORK 4 SOLUTIONS
MATT RATHBUN
MATH 310, SECTION 3
ABSTRACT ALGEBRA
3.1 #1, 5, 6, 12, 14, 18, 27, 29; 3.2 #3, 4, 12, 13, 19
Section 3.1.
1.
(a) Closure of addition, because 1 + 3 = 4, which is not odd.
(b) The existence of additive inverses, because 1
HOMEWORK 3 SOLUTIONS
MATT RATHBUN
MATH 310, SECTION 3
ABSTRACT ALGEBRA
Appendix D #5, 8, 16, 17; 2.2 #2, 6, 10; 2.3 #3, 4, 6
Appendix D.
5.
(a)
Reexivity: Let (x, y ) be any point in the plane. Then (x, y ) (x, y ) because x = x.
Symmetry: Suppose (x, y
NaKyijah White
Lesson 1.08. Laboratory: Nakyijah
Introduction
Write a short (2-3 sentence) explanation about this lab.
Materials
Rocks, 2 Aluminum Pans, Syringe Thing, ruler, water sun , freezer
Hypothesis
I think the water will melt and it will fall to a
Math | Extended Problems | Number Properties
Name:
Date:
Extended Problems
Number Properties
Solve the problems. When you have finished, submit this assignment to your teacher by the due date for full
credit.
Total score: _ of 15 points
(Score for Questio
MATH 310, HOMEWORK 10
1. Recall that the eld of complex numbers C is equal to R2 with the usual addition
of vectors and the following celestial multiplication rule:
(r, )(r , ) = (rr , + )
(in polar coordinates)
(a, b)(c, d) = (ac bd, ad + bc)
(in rectang
MATH 310, HOMEWORK 6
1. (20 points) We say
in Zn .
a exists in Zn if the equation x2 = a has at least one solution
(a) For which odd prime numbers p does
2 exist in Zp ?
(b) For which odd prime numbers p does 1 exist in Zp ?
Hint: You will probably need t
MATH 310, FINAL REVIEW SOLUTIONS
List of Denitions:
A function f : X Y is a subset F X Y such that if (a, b) F and (a, c) F ,
then b = c. Notation: (a, b) F f (a) = b.
A function f : X Y is injective if for any a, b X such that a = b, we have
f (a) = f (b
MATH 310, HOMEWORK 7 SOLUTIONS
1. (20 points) Suppose we distribute m objects among n bins in some way. If a bin
contains more than one object, we say it is overloaded. Write a careful proof for each
of the following statements:
(a) If m > n, prove that a
MATH 310, HOMEWORK 8 SOLUTIONS
1. Let a = (a0 , a1 , a2 , . . . ) be an innite sequence of real numbers. Consider the
operator E which multiplies the k -th term by k and then shifts the whole sequence one
step to the left:
E (a) = E (a0 , a1 , a2 , . . .
MATH 310, HOMEWORK 9 SOLUTIONS
1. Recall that an n-pointed star is dened to be a diagram where we draw an arrow
from x to x + k for each x Zn . We know that there are exactly n dierent n-pointed
stars. Also, recall that a piece of a star is a part of the
MATH 310, HOMEWORK 10 SOLUTIONS
1. Recall that the eld of complex numbers C is equal to R2 with the usual addition
of vectors and the following celestial multiplication rule:
(r, )(r , ) = (rr , + )
(in polar coordinates)
(a, b)(c, d) = (ac bd, ad + bc)
(
MATH 310, HOMEWORK 11 SOLUTIONS
1. If S is a commutative ring with identity, let S 2 = cfw_(a, b) | a, b S with the following
addition and multiplication:
(a, b) + (c, d) = (a + c, b + d)
(a, b)(c, d) = (ac bd, ad + bc)
where a, b, c, d S . Recall that S
MATH 310: REVIEW PROBLEMS
1. (a) If m is a multiple of both 10 and 6, then is m a multiple of 60?
(b) Complete the sentence: If m is a multiple of both a and b, then m is a
multiple of
.
(c) If m is a multiple of both a and b, then what additional conditi
MATH 310: REVIEW PROBLEMS
1. (a) If m is a multiple of both 10 and 6, then is m a multiple of 60?
(b) Complete the sentence: If m is a multiple of both a and b, then m is a
multiple of
.
(c) If m is a multiple of both a and b, then what additional conditi