Instructor |
Dragos Oprea
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Lectures: |
MWF 10:00-10:50AM, PETER 103 |
Course Assistants |
Artem Mavrin
- Discussion 1: Monday, 7:00-7:50PM in
B412
- Discussion 2: Monday, 5-5:50p in APM 7421
- Office: 6446
- Office
hour: Monday 3-4, 6-7; Tuesday 3-5
- Email:
amavrin at
ucsd dot edu.
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Course Content |
Elementary number theory. Topics include
unique factorization, residue systems,
congruences, primitive roots, reciprocity laws, quadratic forms, Diophantine equations. |
Prerequisities: |
Math 31CH or Math 109, or consent of the instructor.
|
Grade Breakdown | The grade is computed as the following
weighed average: - Homework 20%, Midterm I 20%, Midterm II 20%,
Final Exam 40%.
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Textbook: | W. J. LeVeque, Fundamentals of Number
Theory. |
Readings | Reading the sections of the textbook
corresponding to
the assigned homework exercises is considered part of the homework
assignment. You are responsible for material in the assigned reading
whether or not it is discussed in the lecture. It will be expected that
you read the assigned material in advance of each lecture. |
Homework |
Homework problems will be assigned on the
course
homework
page. There will be 7 problem sets, due Wednesdays at
4:30PM
in
the TA's mailbox. You may work together with your classmates
on your
homework
and/or ask the TA (or myself) for
help on assigned homework problems. However, the work you turn in must be
your own. No late homework assignments will be accepted.
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Midterm Exams | There will be two midterm exams given
on October 24 and November 21. There will be no makeup
exams. Regrading policy: graded exams will be handed back in
section.
Regrading is not possible after the exam leaves
the room. |
Final Exam | The final examination will be held on
Friday, December 19, 8AM-11AM. There is no
make up final
examination. It is your responsability
to ensure that you do not have a schedule conflict during the final
examination; you should not enroll in this class if you cannot
sit for the final examination at its scheduled time. |
Announcements and
Dates |
- Friday, October 3: First lecture
- Friday, October 24: Midterm I
- Tuesday, November 11: Veterans' Day.
- Friday, November 21: Midterm II
- Thursday-Friday, November 27-28:
Thanksgiving. No class.
- Friday, December 12: Last Lecture
- Friday, December 19: FINAL EXAM,
8AM-11AM.
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Exams |
- Preparation for Midterm 1:
- Preparation for Midterm 2:
- Preparation for Final Exam:
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Lecture Summaries | -
Lecture 1: Introduction and motivation. Examples of problems addressed
by number theory.
- Lecture 2: Algebraic structures:
monoids, groups, rings, domains, fields. Some examples. Gaussian integers,
Eisenstein integers, quadratic integers.
- Lecture 3:
Subgroups. Subrings. Subfields. Homomorphism and isomomorphism. Ordered
rings and well-ordering. Definition of the integers.
- Lecture
4: Division theorem and proof. Divisibility and its properties.
Greatest common divisor.
- Lecture 5: Existence and
uniqueness of the greatest common divisor. The gcd is expressed as a
linear combination of the two numbers. Existence and uniqueness of the
prime factorization.
- Lecture
6: There are infinitely many primes. Computing the gcd and lcm from the
prime factorization. Computing the gcd via Euclid's algorithm.
- Lecture 7: Proof of Euclid's algorithm. Solving linear
diophantine equations via Euclid's algorithm. General solution to linear
diophantine equations.
- Lecture 8: Euclidean algorithm in
arbitrary domains implies existence of gcd, solvability of linear
diophantine equations. Euclidean norms. Norms over quadratic integers.
Gaussian integers form an Euclidean domain.
- Lecture 9:
Irreducible elements. Prime elements. Prime implies irreducible. Example
of irreducible but not prime element in a domain. Unique factorization
domains. In UFD, primes and irreducible coincide.
- Lecture
10: Euclidean domains are unique factorization domains. Gaussian
integers form a UFD.
- Lecture 11: Congruences. Congruence
is an equivalence relation. The ring of integers modulo m and its
operations. Examples, multiplication table. Investigation of
when Z_m is a domain, what the units are etc.
- Lecture 12:
The units in Z_m. Finding inverses of units via Euclidean algorithm.
Euler's function. Multiplicative functions.
- Lecture 13:
Examples of multplicative functions: number of divisors, sum of divisors.
The Euler function is
multiplicative. Proof via the Chinese remainder theorem.
-
Lecture 14: Linear systems of congruences. Reduction to "normal form",
finding solutions via the Chinese remainder theorem. Examples.
- Lecture 15: General solutions to linear systems of congruences
whose moduli are not coprime. Examples. Motivation for higher degree
congruences and the need for further tools.
- Lecture 16:
Complete and reduced residue systems. Fermat and Euler's theorems. Proof.
Examples. Reduction of degree of congruences via Fermat and Euler.
- Lecture 17: Higher degree polynomial
congruences. Lagrange's theorem about the number of solutions.
Applications of Lagrange: the polynomial x^{p-1}-1 and Wilson's theorem;
the polynomial x^{d}-1.
- Lecture 18: Solving quadratic
congruences. Quadratic residues and non-residues. Legendre symbol and
properties.
- Lecture 19: Higher degree polynomial
congruences modulo powers of primes. Taylor's theorem and constructing
solutions inductively. Singular and non-singular solutions.
-
Lecture 20: More on Taylor's theorem and examples. Midterm review.
- Lecture 21: Primitive roots. Examples. Order of
elements modulo m. Order divides the Euler function. Order of
powers of elements in terms of order of the original element.
- Lecture 22: Existence of primitive roots modulo a prime. There
are exactly \phi(p-1) primitive roots modulo a prime.
Finding all primitive roots once we know a single one.
Generalizations modulo powers of a prime.
- Lecture 23: Existence of primitive roots modulo squares of
primes. Preparations for the existence of primitive roots modulo powers of
odd primes.
- Lecture 24: Existence of primitive roots modulo powers of odd
primes. There are no primitive roots modulo powers of 2. Generating the
units mod 2^e via signed powers of 5.
- Lecture 25: Existence of primitive roots modulo 2p^e.
Non existence of primitive roots for the remaining moduli. Examples.
- Lecture 26: Applications of primitive roots. Solving
exponential congruences using primitive roots. Listing quadratic, cubic
residues using primitive roots.
- Lecture 27: Solving the congruence x^n=a in Z_m using
primitive roots. Number of
solutions. Examples.
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