MATH 235: Assignment 5 Solutions
1.1: Writing N = n0 + 10n1 + . . . + 10k nk , we can reduce
fact that 10 is congruent with 1 mod 3, we conclude that
mod 3 and using that
N n0 + 1 n1 + 12 n2 + . . . + 1k nk .
Clearly then N 0 mod 3 i the sum of its digits
Math 235 (Fall 2009): Assignment 10 Solutions
1.1: 28 = 4 7. Since gcd(4, 7) = 1, using the Euclidean Algorithm (or simply by
guessing), we nd that 1 = 4 2 + 7 (1). We dene e1 := 1 4 2 = 7 and
e2 := 1 7 (1) = 8.
It is easy to check (one by one) that the s
ASSIGNMENT 5 - MATH235, FALL 2007 Submit by 16:00, Monday, October 15 (use the designated mailbox in Burnside Hall, 10th floor).
1. To check if you had multiplied correctly two large numbers A and B, A B = C, you can make the following check: sum t
Math 235 (Fall 2009): Assignment 2 Solutions
1: Let us prove by induction that f (0, n) = n(n+1) . It is clearly true in the case
2
n = 0.
Assume it holds for n. Consider the diagonals in the second list (e.g. the diagonal
6, 7, 8, 9). Notice that, by con
MATH 235 Assignment 7 Solutions 1.1: x2 3 is quadratic and hence is irreducible over Q/R i it no roots in has the corresponding elds. But x2 3 = (x 3)(x + 3) and since 3 lies in R but not Q we conclude x2 3 is irreducible in Q[x] but not in R[x]. 1.2: As
MAT235 Assignment 9 Solutions
1: As noted in the hint, we rst consider the total number of squares in F , where
|F| = q n for some prime q and positive integer n. Clearly the set S := cfw_x2 |x F
runs through all such square, we consider the cardinality
MAT235 Assignment 11 Solutions
5: Clearly the identity lies in the intersection of 2 subgroups. If a and b also lie in
the intersection, then so does a1 and ab since H1 and H2 are subgroups and hence
closed under inversion and multiplication. Therefore so
Algebra I, MATH 235
Fall 2015
Instructor: Prof. Eyal Goren
Office: Burnside Hall 1108
Office hours: MWF, 14:30 - 15:30
Lecture hours: MWF, 13:35 - 14:25, Maass 112
TA Office hours and tutorial sessions:
Bruno Joyal, Monday 15:00 - 16:00, Friday, 11:00 to
Math 235 (Fall 2009): Assignment 8 Solutions
4.a: It is easy to see the function is bijective. The remaining part is computational.
4.b: We can easily check that f (xi) = f (x)f (i), f (xj) = f (x)f (j) and f (xk) =
f (x)f (k) for any x R. By denition of
Math 235 (Fall 2009): Assignment 4 Solutions
1.1: Let m = [a, b] and k be an integer such that a|k and b|k. Let q and r be
integers such that k = mq + r and 0 r < m. Since a|k and a|m, a|r. Similarly,
b|r. But, by denition of m, r has to be zero (why?). H
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cfw_\H
Number Theory - Factors, GCD, and Primes
1
Introduction
Number theory is the branch of mathematics that deal with properties of integers. However, it has a
very close tie with computer science. The most notable applications of number theory is in the eld
Fundamental Theorem of Arithmetic and Divisibility Review Mini Lecture
Here we will provide a proof of the Fundamental Theorem of Arithmetic (about prime factorizations).
Before we get to that, please permit me to review and summarize some divisibility fa
THE GAUSSIAN INTEGERS
KEITH CONRAD
Since the work of Gauss, number theorists have been interested in analogues of Z where
concepts from arithmetic can also be developed. The example we will look at in this handout
is the Gaussian integers:
Z[i] = cfw_a +
Math 455.1 April 4, 2009
Fermats Little Theorem
For the RSA encryption system, we shall need the following result
Theorem 1 (Fermats Little Theorem). Let p be a prime. Then for each integer
a not divisible by p,
ap1 1 (mod p).
Proof. Let a be an integer f
SUMS OF TWO SQUARES
PETE L. CLARK
1. Sums of two squares
We would now like to use the rather algebraic theory we have developed to solve a
purely Diophantine problem: which integers n are the sum of two squares?
The relevance of the Gaussian integers is n
Solution 1:
By the rst homomorphism theorem we know that
R
=
Z
ker f .
Recall ker f is an ideal in Z. Since Z is a PID we see that ker f = (n) for some integer n.
Case 1: Suppose n = 0.
This implies
R
=
Case 2: Suppose n = 0.
Then (n) = nZ. Thus
R
=
Z.
=
Final Examination
December 17, 1996
189-235A
1.
Say whether the following statements are true or false. You do not need to provide
justi cation.
a The ring Z x of polynomials with integer coe cients is euclidean.
b The ring R x of polynomials with real co
6
Euclidean Domains, PIDs and UFDs
We now consider three classes of integral domains that have additional structure, allowing us to say a good deal more about there algebraic properties.
These are unique factorization domains, or UFDs, which allow unique
Important theorems about ring homomorphisms and ideals.
In the following propositions, the letters R and S will always denote rings.
1. Suppose that : R S is a ring homomorphism. Then Ker() is an ideal in the ring R
and Im() is a subring of the ring S . (
To begin with we will need to following facts.
Fact 1:
Let n = pa1 1 . . . pakk be the decomposition of the number n into primes p1 , . . . , pk . Then
a b (mod n)
if and only if
a b (mod pa1 1 )
.
.
(1)
k
a b (mod pak ).
Proof:
Suppose a b (mod n). Then
Math 235 (Fall 2012)
Assignment 1 solutions
Luiz Kazuo Takei
September 25, 2012
Exercise 1
a) x3 + 3x + 1
In this case, the discriminant is
12 + 4
33
>0
27
which means there is only one real solution.
b) x3 3x + 1
In this case, the discriminant is
12 + 4
MATH 235 ALGEBRA I
SOLUTIONS TO ASSIGNMENT 10
Page 111 Exercise 8
Let be a permutation of the elements cfw_1, 2, . . . , n i.e. an element of Sn . We want to show
that can be written as a product (i.e. composition) of transpositions (i.e. 2-cycles). Every
SOLUTIONS TO MIDTERM EXAM MATH 235, FALL 2015
October 27, 2015
Time: 90 Minutes.
Instructions: Each question is worth 20 points. Answer 5 of the following questions. Answer a
sixth for bonus points. The maximal grade for this exam is capped at 100.
(1) Fi
Algebra I, MATH 235
http:/www.math.mcgill.ca/goren/MATH235.2015/MATH235.html
Algebra I, MATH 235
Fall 2015
Instructor: Prof. Eyal Goren
Office: Burnside Hall 1108
Office hours: MWF, 14:30 - 15:30
Lecture hours: MWF, 13:35 - 14:25, Maass 112
TA Office hour