Math 104A: Number Theory
Fall 2012

Instructor's Office: 7250 APM
Instructor's Office Hours: 10:00-11:00am MWF or by appointment
Lecture Time: 9:00-9:50am MWF
Lecture Room: B412 APM
TA: Tristan Sandler
TA's Email: tjsandle (at) ucsd.edu
TA's Office: 6351 APM
TA's Office Hours: 11:00am-1:00pm Tu
Discussion Time: 8:00-8:50am Tu
Discussion Room: B412APM
Final Exam Time: Wednesday, 12/12/2012, 8:00-10:59 am
Final Exam Room: To Be Announced

Lectures:

9/28: Syllabus, administrivia, the set Z of integers, prime numbers, statements of the Prime Number Theorem and the Unique Factorization Theorem, binary operations. (Sections 1.1, 1.2)
10/1: Group axioms, abelian groups, subgroups, finite cyclic groups, the order of a group, group homomorphisms/isomorphisms. (Section 1.2)
10/3: Ring axioms, examples of rings, domains, ring homomorphisms/isomorphisms, definition of a field. (Section 1.2)
10/5: Proof that every field is a domain, ordered rings and the definition of <, well-ordering, characterization of Z as the unique ordered domain whose positive elements are well-ordered, first example of well-ordering's use in a proof. (Sections 1.2, 1.3)
10/8: Proof that there are no rational points on the circle x^2 + y^2 = 6, proof that every nonzero integer has a prime factorization, the Division Theorem, proof that if p is prime and p|ab, then p|a or p|b. (Sections 1.3, 1.4)
10/10: Proof of the Unique Factorization Theorem for Z, Proof that there are infinitely many positive primes, Definition-existence-uniqueness of the greatest common divisor (Sections 1.4, 2.1)
10/12: More on greatest common divisors, the Euclidean Algorithm, the structure of {ax + by : x,y in Z} for a, b fixed. (Section 2.1)
10/15: Definition of units, Euclidean domains. Examples: Z, polynomial rings over fields, Gaussian integers. (Section 2.2)
10/17: Definitions of primes and associates in arbitrary domains, definition of Unique Factorization Domains, statement of the theorem that every Euclidean domain is a UFD, proof that every element of a Euclidean domain has a prime factorization. (Section 2.2)
10/19: Definition of divisibility in an arbitrary domain, definition/characterization of the gcd in an arbitrary Euclidean domain (or even an arbitrary UFD), proof that every Euclidean domain is a UFD. (Section 2.2)
10/22: The linear Diophantine equation ax + by = c, the postage stamp problem, the least common multiple. (Sections 2.3, 2.4)
10/24: Midterm 1.
10/26: Midterm 1 postmortem, anonymous course feedback.
10/29: Equivalence relations, equivalence classes, congruence modulo m, the set of integers modulo m. (Section 3.1)
10/31: Proof that modular addition and multiplication are well defined and give Z_m the structure of a ring. Units in Z_m and characterization of when Z_m is a domain/field. Using the Euclidean Algorithm to find inverses in Z_m. (Section 3.1)
11/2: Complete and reduced residue systems, the unit group U_m of Z_m, Euler's totient/phi-function, and explicit formula for the phi-function. (Sections 3.1, 3.2)
11/5: More about the phi-function, Euler's Theorem, Fermat's Little Theorem, the definition of ord_m(a) for (m,a) = 1, properties of orders. (Section 3.2)
11/7: Linear congruences, systems of linear congruences and the Chinese Remainder Theorem, the analog of CRT with not-coprime moduli. (Section 3.3)
11/9: More on the Chinese Remainder Theorem: Solving explicit systems of linear congruences, introduction to the polynomial ring Z_m[x]. (Sections 3.3, 3.4)
11/14: The ring Z_p[x] when p is prime: bound on the number of roots of a polynomial, factorization of x^p - x, Wilson's Theorem. (Section 3.4)
11/16: Using Taylor's Theorem to solve f(x) = 0 (mod p^e), square roots in Z_p, Euler's Criterion. (Section 3.4)
11/19: Midterm 2.
11/26: Midterm 2 postmortem, Legendre symbols (a/p), quadratic residues/non-residues. (Section 3.4)
11/28: Primitive roots, relationship between the orders of g and g^n, proof that U_p has primitive roots for p > 0 prime. (Section 4.1)
11/30: Structure theorem relating ord_p(a) and ord_{p^n}(a). Applications: U_{p^n} has a primitive root for p > 2, structure of U_{2^n}. (Section 4.1)
12/3: The structure of U_m for arbitrary m, proof that U_m has a primitive root iff m = 2, 4, p^n, or 2p^n for p an odd prime, indices and index vectors. (Sections 4.1, 4.2)
12/5: nth power residues in U_m, criterion for existence/number of nth power residues in U_{p^e}, p prime, reduction of quadratic residues in U_m to quadratic residues in U_p, p > 2 prime. (Sections 4.3, 5.1)
12/7: Properties of the Legendre symbol (a/p), Gauss's Lemma and a formula for (2/p), statement of the Quadratic Reciprocity Theorem. (Section 5.2)

Homework Assignments:

Homework 1, Due 10/5/2012.
Homework 2, Due 10/12/2012.
Homework 3, Due 10/19/2012.
Homework 4, Due 11/2/2012.
Homework 5, Due 11/9/2012.
Homework 6, Due 11/16/2012.
Homework 7, Due 12/3/2012.

Midterms: