Here you will see information on what was covered in class, upcoming quizzes, and solutions to exams

CHECK HERE EVERYDAY !

COURSE PAGE WITH SYLLABUS, GRADING SCHEME, ETC.,

SYLLABUS

Will try to stick to this syllabus as much as possible.

NOTES AND EXAM SOLUTIONS FROM FALL 2022.

PLEASE READ ALL INSTRUCTIONS ABOVE AND KEEP UP WITH CLASS.

THURSDAY OFFICE HOURS ON ZOOM.

10/11/2024 FRIDAY

Notes on Unique Factorization (Updated -- please reload!).

On wednesday we proved that in a PID you have factorization into irreducibles.

More HW3 problems: (HW3 DUE WEDNESDAY)

5. Show that an element u = a+ib in Z[i] is a unit iff a^2+b^2 = 1. Find all the units in this ring.

6. Show that a+ib in Z[i] is irreducible if a^2+b^2 is a prime number.

7. Let R be a Euclidean domain that is not a field. Show that there is a in R such that every coset in R/(a) is of the form (a)+r where r is either 0 or a unit of R.

8. A GCD-domain is an integral domain in which any two elements a, b have a GCD: an element g such that g divides a and b, AND if d divides a and b then d divides g. Show that irreducible elements in any GCD domain are also prime elements.

9. Show that an integral domain that is Noetherian (see notes) has factorization into irreducibles (not necessarily unique).

10. Show that Noetherian domains which have GCD property are UFD's.

11. Z[x] is known to be Noetherian (see notes). Show that Z[x] is also a GCD domain, and hence a UFD (note that it is neither Euclidean nor a PID).

Notes on Unique Factorization.

Today we talked about Euclidean domains, PIDs, and primes and irreducibles.

Some HW3 problems:

1. Show that Z[x] is not a PID, where Z is the ring of integers.

2. We showed in class that Z[x] cannot be a Euclidean domain because Z is not a field. Explain why this makes Euclidean algorithm not work.

3. Show that Q[x]/(f(x)) is a field whenever f(x) is an irreducible polynomial in the field of rational numbers Q.

4. If f(x) = x^2+x+1 in (3) find the image of x^3+2 mod f(x) and its inverse.

Test 1 Solutions

test 1 Problems

Test I will be on wednesday Sep 23. It will cover upto chapter 6.

REVIEW IN CLASS ON MONDAY.

9/23/2024 MONDAY

Problem 16: Primes in Pythagorean Triples.

For the test please study sample problems, class notes, homework problems, old tests (from fall 2022). Note that test 1 in fall 2022 covered a few more topics.

Please study these sample problems for Test 1.

We will go over as many of them monday as possible.

For the test please study class notes, homework problems, old tests (from fall 2022). Note that test 1 in fall 2022 covered a few more topics.

9/11/2024 WEDNESDAY

We started talking about GCD.

Notes for today and next Monday

HW2 problems: Problems 5.1 and 5.3.

Please do Problem 2.4 from book using Gaussian integers

Problems 3.3, 3.4 and 3.5 from Chapter 3. In 3.5 part (f) is hard-- but you can give it a try just for fun.

9/9/2024 MONDAY

We finished the parametrization of Pythagorean triples using unit circle. Talked about the trigonometric proof and the proof using Gaussian integers.

Pythagorean Triples(Revised from fall 22)

HW2 problems: Please do Problem 2.4 from book using Gaussian integers

Problems 3.3, 3.4 and 3.5 from Chapter 3. In 3.5 part (f) is hard-- but you can give it a try just for fun.

9/4/2024 WEDNESDAY

We described the parametrization of Pythagorean triples using unit circle. Please read the notes because I have added some things to what we talked about in class.

Pythagorean Triples(Revised from fall 22)

Please try Problem 2.8 from book (just for fun!)

8/28/2024 WEDNESDAY

We described the parametrization of Pythagorean triples. Please read the notes because I have added some things to what we talked about in class.

Pythagorean Triples(Revised from fall 22)

Homework problems: Problems 2.5 and 2.6 from book. All of the homework problems given so far are due next wednesday.

8/27/2024 TUESDAY

Yesterday we finished the intro and started on Pythagorean triples. Please read the notes because I have added some things to what we talked about in class.

Updated notes (please refresh!)

INTRODUCTION TO NUMBER THEORY (Revised from fall 22)

Homework problems: Problems 2.1a and 2.2 from book: 2.1. (a) We showed that in any primitive Pythagorean triple (a, b, c), either a or b is even. Use the same sort of argument to show that either a or b must be a multiple of 3.

2.2. A nonzero integer d is said to divide an integer m if m = dk for some number k. Show that if d divides both m and n, then d also divides m-n and m + n.

8/21/2024 WEDNESDAY

HW problems for today: (see notes below for what we did in class).

1. We saw that product of numbers of form 4n+1 is also of form 4n+1. Same is not always true for numbers of form 4n-1. When does product of such numbers also result in a number of form 4n-1?

2. (problem 12.2, part 1) Use the ideas from the case of 4n-1 (or 4n+3) to show that there are infinitely many primes of form 6n+5.

3. Try problems 1.3 and 1.4 after chapter 1 of the text.

If you don't have the text you can find the chapter on Prof. Silverman's website.

More hw will be assigned monday and they are all due on monday following week.

8/19/2024 MONDAY

Notes from today: INTRODUCTION TO NUMBER THEORY (Revised from fall 22)

8/16/2024 FRIDAY

Please study following to prepare for class: INTRODUCTION TO TEXT BOOK