×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

### Create a StudySoup account

#### Be part of our community, it's free to join!

or

##### By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

34

0

2

# INTRO MDRN ALG MATH 411

UW
GPA 3.76

Staff

These notes were just uploaded, and will be ready to view shortly.

Purchase these notes here, or revisit this page.

Either way, we'll remind you when they're ready :)

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

### Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

COURSE
PROF.
Staff
TYPE
Study Guide
PAGES
2
WORDS
KARMA
50 ?

## Popular in Mathematics (M)

This 2 page Study Guide was uploaded by Addison Beer on Wednesday September 9, 2015. The Study Guide belongs to MATH 411 at University of Washington taught by Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/192109/math-411-university-of-washington in Mathematics (M) at University of Washington.

×

## Reviews for INTRO MDRN ALG

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/09/15
Mathematics 41 1 11 December 2002 Final exam preview Instructions As always in this course clarity of exposition is as important as correctness of mathematics 1 N E Recall that a Gaussian integer is a complex number of the form a bi where a and b are integers The Gaussian integers form a ring Zi in which the number 3 has the following special property Given any two Gaussian integers r and s if 3 divides the product rs then 3 divides either r or s or both Using this fact prove the following theorem Given any Gaussian integers r1 r2 divides at least one of the factors ri rquot if3 divides the product r1 in then 3 a Use the Euclidean algorithm to nd an integer solution to the equation 7x 37y l b Explain how to use the solution of part a to nd a solution to the congruence 7x El mod 37 c Use part b to explain why 7 is a unit in the ring Z37 Given a Gaussian integer t that is neither zero nor a unit in Zi a factorization t rs in Zi is nontrivial if neither r nor 3 is a unit a List the units in Zi indicating for each one what its multiplicative inverse is No proof is necessary b Provide a nontrivial factorization of 53 in Zi Explain why your factorization is nontriv ial c Suppose that p is a prime number in Z that has no nontrivial factorizations in Zi Prove that the equation x2 y2 p has no integer solutions Mathematics 41 1 Palmieri 4 Suppose that R is a ring with additive identity 0 An element of r of R is called a zero divisor V39 9 7 if r is nonzero and if there is a nonzero element 3 of R such that rs 0 a Describe a zerodivisor in Z 10 and explain why it is one b Suppose that m is an integer greater than 2 that is not a prime Describe a zerodivisor in Zm and explain why it is one c Suppose that p is a prime number Recall that if p divides a product ab of two integers then it divides at least one of a and b Use this to prove that there are no zerodivisors in Z p One can construct a ringR with six elements R 61 b c d ef with multiplication table as follows a Which element of R is the multiplicative identity Why b Which elements of R are units Why c Is R a eld Why or why not d Can the multiplication table above be the multiplication table of Z 6 with the six ele ments of Z6 being assigned somehow the names a b c d e and f In this problem 1 represents the Euler phifunction and m is a positive integer a De ne b Suppose m pe for some prime number p and positive integer 6 State a formula for in terms of p and e and prove that the formula is correct c State Euler s theorem Make sure that all of the terms occurring are clearly identi ed and any restrictions on them are described d Use Euler s theorem to nd the smallest positive integer c such that 5773 E 6 mod 121 Explain what you are doing in your calculation pointing out in particular how Euler s theorem is used When was Herman Melville born In what city

×

×

×

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Steve Martinelli UC Los Angeles

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Amaris Trozzo George Washington University

#### "I made \$350 in just two days after posting my first study guide."

Bentley McCaw University of Florida

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Parker Thompson 500 Startups

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.