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 + . . . +
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,
ASSIGNMENT 1 - MATH235, FALL 2009
Submit by 16:00, Monday, September 14 (use the designated mailbox in Burnside Hall, 10th
floor).
1. Calculate the following intersection and union of sets (provide sh
ASSIGNMENT 8 - MATH235, FALL 2009
Submit by 16:00, Monday, November 9
Do the following questions from pages 86-88 in the course notes.
Questions (4), (7), (16), (17), (18).
ASSIGNMENT 5 - MATH235, FALL 2009
Submit by 16:00, Tuesday, October 13 (use the designated mailbox in Burnside Hall, 10th
floor).
1. Given an integer N we write N in decimal expansion as N = nk nk1 .
ASSIGNMENT 3 - MATH 235, FALL 2009
Submit by 16:00, Monday, September 28 (use the designated mailbox in Burnside Hall, 10th
floor).
1. Find the quotient and remainder when a is divided by b:
(1)
(2)
(
ASSIGNMENT 10 - MATH235, FALL 2009
Submit by 16:00, Monday, November 23
1. Solve the following equations using the Chinese Remainder Theorem. Find all the solutions.
(1) x 3 x + 8 modulo 28.
(2) x 2 9
MATH 235, Sample Final Solutions
Solutions by Jonathan Boretsky, Ayoub El-Hanchi, Samuel Fisher. I have made some slight
changes to their original phrasing, and added in some additional remarks. This
ASSIGNMENT 11 - MATH235, FALL 2009
DO NOT SUBMIT. THESE ARE SUGGESTED EXERCISES THAT MATCH THE MATERIAL COVERED AT PRESENT IN THE COURSE. SIMILAR QUESTIONS OFTEN APPEAR
ON THE FINAL EXAM.
Solve questi
ASSIGNMENT 9 - MATH235, FALL 2009
Submit by 16:00, Monday, November 16
1. Let F be a finite field. Prove that if A and B are two non-zero elements of F that are not squares
then A/B is a square. (Sugg
ASSIGNMENT 7 - MATH235, FALL 2009
Submit by 16:00, Monday, November 2
1. Is the given polynomial irreducible:
(1) x 2 3 in Q[x]? In R[x]?
(2) x 2 + x 2 in F3 [x]? In F7 [x]?
2. Find the rational roots
ASSIGNMENT 2 - MATH235, FALL 2009
Submit by 16:00, Monday, September 21 (use the designated mailbox in Burnside Hall, 10th
floor).
1. Consider N N as a rectangular array:
(0, 0)
(1, 0)
(2, 0)
(3, 0)
.
Assignment 3 Solutions
MATH 235
From Abstract Algebra, by T.W. Judson, Section 2.3
Exercise 3: We want to prove that n! > 3n for every integer n 6 by induction.
Base step: The claim holds for n = 6,
Weekly HW Assignments for Math 235.
WARNING: THIS PAGE WILL BE UPDATED AND ADJUSTED REGULARLY. DONT START HW UNTIL REVISITING IT.
The exercises are mostly drawn from the text. There are hints an
MATH 235, Midterm (Version 2) Solutions
1. Solve the equation 15x 6 = 0 in the ring Z/49Z.
Solution. Applying the Euclidean algorithm, we get
49 = 3 15 + 4
15 = 3 4 + 3
4 = 13+1
We conclude that gcd(4
MATH 235, Sample Midterm
Duration: 40 Minutes
Name:
Student number:
Signature:
Every question is worth 10 marks, good for a total of 40 marks on the entire midterm.
Start with the questions you are mo
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 conclud
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
run
MAT235 Assignment 3 Solutions
1.1:
1.2:
1.3:
1.4:
1.5:
1.6:
302 = 15 19 + 17. the quotient is 15 and the remainder is 17.
302 = 16 19 + 2. the quotient is -16 and the remainder is 2.
0 = 0 19 + 0. q =
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
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 (
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,
Abstract Algebra
Theory and Applications
Abstract Algebra
Theory and Applications
Thomas W. Judson
Stephen F. Austin State University
Sage Exercises for Abstract Algebra
Robert A. Beezer
University of
MATH 235
Sets and Functions
1
Magid Sabbagh
Injectivity and Surjectivity
Definition: A function f : A B is a subset f A B such that for every a A, there exists a
unique b B such that (a, b) f .
Defini
Assignment 11, MATH 235, Fall 2015
Eyal Goren
December 3, 2015
Do not submit this assignment.
Answer the following questions from pages 106-107: (11), (12), (13), (14).
In addition, solve the followin