Fall 2012

**Instructor:** Brendon Rhoades

**Instructor's Email:** bprhoades (at) math.ucsd.edu

**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

** Syllabus:** Please read carefully.

** 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.

Running List of Definitions/Theorems/Concepts

** Midterms: **

Midterm 1
Solutions and Comments

Score distribution: 94, 92, 89, 85, 80, 79, 79, 78, 78, 77, 76,
75, 73, 69, 68, 67, 67, 58, 57, 50, 50, 48

Midterm 2
Solutions and Comments

Score distribution: 100, 99, 98, 94, 93, 83, 81, 80, 80, 80, 78,
76, 75, 75, 73, 71, 65, 65, 65, 63, 60, 53

Final exam score distribution:
162, 162, 160, 158, 146, 142, 126, 120, 119, 118, 109,
107, 103, 100, 100, 94, 91, 86, 84, 72, 53, 46