HOMEWORK 10
DUE APRIL 29TH, 2010
1. Let n > 1 be an integer. Suppose that b is a positive integer such that b is
a square modulo bn 1. Conclude that there are numbers a11 , a12 , a22 such that
a22 = bn 1, b = a11 a22 a2 , and a11 > 0.
12
2. Let b be a num
HOMEWORK 7
DUE APRIL 6TH, 2010
1. Section 5.1: 4ab
2. Section 5.2: 1, 3
3. Suppose that A and B are n m matrices with A B modulo m Let C be a
m p matrix. Show that AC BC modulo m.
4. a) Compute the adjoint matrix of
1
A = 1
1
2
2
4
3
5
6
b) Use your answe
HOMEWORK 8
DUE APRIL 13TH, 2010
1. Section 3.5: 1, 4, 5, 9, 10, 11
2. a) Give 3 examples of reduced Pythagorean triples where one of the numbers is
at least 100.
b) Show that there are no solutions to Fermats equation xn + y n = z n if 4|n unless
one of x
HOMEWORK 9
DUE APRIL 20TH, 2010
1. Section 3.6: 11, 12
2. Section 5.5: 2, 3
3. a) Let f (x, y, z ) be a ternary quadratic form. How do you identify f (x, y, z ) with
a symmetric, 3 3 matrix?
b) Write the entries of this matrix as rij and call the matrix R
HOMEWORK 6
DUE MARCH 16TH OR MARCH 23RD, 2010
1. Section 3.1: 11, 16.
2. Section 3.2: 4, 7, 13
3. Suppose p is a prime number and (a, p) = 1. When is ax2 + bx + c 0(p)
solvable. How many solutions does it have?
4. Prove that there are innitely many primes
HOMEWORK 5
DUE MARCH 9TH, 2010
Notes:
1. Section 2.8: 2, 4, 8, 9, 10, 11, 13
EXTRA CREDIT: Section 2.8: 36
2. In this problem we will investigate primitive roots mod powers of 2.
a. Show that there are primitive roots mod 2 and mod 4, but not mod 8.
b. No
HOMEWORK 1
DUE TUESDAY FEBRUARY 2, 2010
1. Let R and S be two commutative rings. Dene R S to be the set of ordered
pairs (r, s) where r R and s S . Dene addition and multiplication of the
ordered pairs component wise. That is (r1 , s1 ) + (r2 , s2 ) = (r1
HOMEWORK 2
DUE TUESDAY FEBRUARY 9, 2010
Change: Please do two of numbers 1, 2, 3, and then two of the problems from
number 4 and three from number 5.
1. Section 1.4 : 4
2. Show that there is a ring homomorphism Z/mZ Z/dZ if and only if d|m.
3. Let f (x) b
HOMEWORK 3
DUE TUESDAY FEBRUARY 16TH, 2010
Notes: Please do numbers 1, 2, 5 and then either problem 3 or problem 4.
1. Section 2.1 : 18
2. Section 2.2 : 3, 7, 14
3. Section 2.1 : 59, 60
4. Section 2.3 : 19, 20
5. We saw in class that if gcd(m1 , m2 ) = 1
HOMEWORK 4
DUE MARCH 2ND, 2010
Notes: This assignment is longer than usual because you have more time than
usual. I wouldnt want you to be bored.
1. Section 2.3: 14, 30.
2. Section 2.4: 5, 6.
3. Section 2.5: 3, 5.
4. Section 2.6: 6, 7, 10.
5. Section 2.7: