Homework 1 (Week 2)
Please write all answers in complete sentences.
Arithmetic in F2
1. Show that the equation x2 + x + 1 = 0 has no solutions in F2 .
2. Show that the equation x3 + x + 1 = 0 has no solutions in F2 .
Some explicit computations with Hammin
Spring 2013
Math 370: Applied Algebra
Prof. Assaf
Solutions to Problem Set 2
1. Using Euclids Algorithm takes a while, but gets the job done with 18 steps. Solving for the
remainder in each equation and substituting back in eventually gives
1 = (4181)4181
Spring 2013
Prof. Assaf
Math 370: Applied Algebra
Solutions to Problem Set 1
3(1)2 1
2
.
2=
2
3n2 n
2 . Then
1. We proceed by induction. The Base Case is for n = 1. In this case we have 1 =
For the Inductive Case, assume that for n 1, 1 + 4 + 7 + + (3n 2)
Spring 2013
Math 370: Applied Algebra
Prof. Assaf
Solutions to Midterm Exam 1
1. Let p N be a prime number. Prove that p is irrational.
Solution : Suppose p is prime and that p is rational. Then p = n for some n, d Z.
d
By the Fundamental Theorem of Arith
Spring 2013
Math 370: Applied Algebra
Prof. Assaf
Solutions to Problem Set 4
1. First suppose that cfw_r1 , r2 , . . . , rm is a complete set of representatives. Then since ri
ri (mod m), the uniqueness of representatives ensures that ri rj (mod m) for
Spring 2013
Math 370: Applied Algebra
Prof. Assaf
Solutions to Problem Set 6
1. By problem 3 on problem set 5, there exists a prime p such that p a = 0 in F. In particular,
the Freshmans Dream applies. Using this p, we have
fp (a + b) = (a + b)p = ap + bp
Spring 2013
Math 370: Applied Algebra
Prof. Assaf
Solutions to Problem Set 10
1. Compute xn (mod x2 + x + 1) in Z/3Z until a pattern appears:
x0 1 (mod x2 + x + 1)
x1 x (mod x2 + x + 1)
x2 x 1 2x + 2 (mod x2 + x + 1)
x3 x(2x + 1) 2x2 + 2x
2(2x + 2) + 2x
Spring 2013
Math 370: Applied Algebra
Prof. Assaf
Solutions to Problem Set 9
1. We rst prove by induction that f (x) can be factored into a product of irreducible polynomials.
If deg(f (x) = 1, then f (x) is irreducible over F. Now assume deg(f (x) > 1 an
Spring 2013
Prof. Assaf
Math 370: Applied Algebra
Solutions to Problem Set 8
1. Let p(x) = x5 and q (x) = x. Then p(x) = q (x) since they have dierent degrees, but, by
Fermats Little Theorem, for any a Z, p(a) = a5 a = q (a) (mod 5).
2. Suppose p(x) Q[x]
Spring 2013
Math 370: Applied Algebra
Prof. Assaf
Solutions to Problem Set 5
1. Since f is a homomorphism, f (0) = 0 and f (1) = 1. For n N, f (n) = f (1) + + f (1) =
1 + + 1 = n, and f (n) = f (n) = n. Therefore f (n) = n for all n Z. For n = 0, n
is a u
Spring 2013
Prof. Assaf
Math 370: Applied Algebra
Solutions to Problem Set 3
1. Since m is not prime, we may write m = ab where 2 a, b m/2. If both a and b are m/2,
then m = m2 /4, so m = 4. Since m = 4, we may assume a < m/2 and b m/2. Write
(m 1)! as a
Spring 2013
Prof. Assaf
Math 370: Applied Algebra
Solutions to Problem Set 11
1. The exponent of U16 is (24 ) = 22 = 4, so every element of U16 is a solution to x4 1. Since
(24 ) = 23 , there are 8 solutions and the group is not cyclic.
2. For U10 , notic
Homework Week 3
Irreducibility
1. Write down all the irreducible polynomials of degree 2 with coefcients in F2 .
2. Write down all the irreducible polynomials of degree 3 with coefcients in F2 .
3. Write down all the irreducible polynomials of degree 4 wi
Homework Week 4
Computing with the (15, 7)-code from class
1. Given the data bit (1, 1, 0, 1, 0, 0, 1), construct the sent word s(x) to which it encodes.
2. Suppose we know that at most 2 errors have occurred in transmission. How would the bit
string (1,
Homework Week 5
Polynomial arithmetic
1. Using the algorithm given in class, determine the greatest common divisor of x3 + 1 and
x4 + 1 over F2 .
More nite elds
2. Suppose n is an integer 2. Consider the collection of symbols 0, 1, . . . , n 1. Given two
Homework Week 8
Orders of elements and Eulers totient function
1. Consider the group Z/mZ of units in the commutative ring Z/mZ discussed on the last
homework.
i) In class, I dened the order of an element g in a (nite) group G, denoted ord(g ), to
be the
Homework Week 9
Groups
1. We know that the exponent of a group divides the order of the group. Consider the group
Z/nZ . The order of Z/nZ is given by Eulers -function, as we studied last week.
i) Compute exp(Z/nZ ) for n = 2, 3, 5, 7, 9, 11, 13.
ii) Show
Homework Week 11
Applications of orders
1. Suppose p > 2 is a prime number, and consider the polynomial x2 + 1 = 0 in Fp .
i) Show that x2 + 1 = 0 has a solution in Fp if and only if there is an element in Z/pZ
that has order 4.
ii) Show that Z/pZ has no
Homework Week 13
On the Chinese remainder theorem
1. Find the smallest positive solution, if any, of
x9
x 17
mod 16
mod 28.
2. Find the smallest positive solution, if any, of
x 10
mod 15
x 17
mod 28.
3. Show that if m = rs with r and s coprime, then x2 1
M idtermL-Math370
Instructions
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M idterm2-Math370
Instructions
There are 130 points
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Mrs. St. Amant
LA 1, Period 4
17 December 2013
My Interesting Title
Indent your first line. Here is where you will write your essay. Be sure to double
space. In order to do that for the whole essay, highlight all of the text you want to