Math 104A (Spring, 2008): Number Theory

Riemann

Gauss

Jacobi

Legendre

Kummer

Dirichlet

Professor: Ron Evans Office: 7349 APM Hours: 3:30 - 4:30 MWF (and after class as needed) Email: revans@ucsd.edu

Teaching Assistant: Daniel Vallieres Office: 6414 APM Hours: Tuesday: 2 pm to 4 pm in APM 6414, and            Thursday: 4 pm to 5 pm in Center Hall 217B Email: dvallieres@math.ucsd.edu

Recent Announcements  

• The Jacobi symbol (c/b) is defined multiplicatively in terms of Legendre symbols. We will restrict b to be an odd integer > 1, while c can be any integer. Here is an example of the definition: (-34/525) = (-34/5)*(-34/5)*(-34/3)*(-34/7). When the denominator of the Jacobi symbol is already prime, then the Jacobi symbol is a Legendre symbol. Note that each Legendre symbol above on the right can itself be split multiplicatively. For example (-34/5) = (-1/5)*(2/5)*(17/5). Note also that while the Legendre symbol tells you if something is a square or not modulo the denominator, the Jacobi symbol need NOT convey this information. For example, (2/9)=(2/3)*(2/3) = (-1)*(-1) = 1, but 2 is NOT a square mod 9.
  
  Let m,n be positive odd integers (not necessarily prime). Then the law of quadratic reciprocity holds for Jacobi symbols, namely (m/n) = -(n/m) when both m,n are -1 mod 4, and (m/n) = (n/m) otherwise. This is the same as the law of quadratic reciprocity for Legendre symbols, when n and m are odd primes.

Homework Homework is due by Friday at 4 pm in TA's box, located on the sixth floor of APM. No late homework will be accepted. Group collaboration is permitted, but solutions should be in your words; verbatim copying is not acceptable. Homework Assignments

Class Schedule Lecture:   Peterson 102    5:00 - 5:50    MWF Discussion section: A01  622364    Center 217B    5:00 - 5:50 pm    Thursday A02  622365    Center 217B    3:00 - 3:50 pm    Thursday

Text and course description Elementary Number Theory  by William Stein Math 104A is an upper division mathematics class devoted to the elementary theory of integers, with applications. Proofs are an essential part of this course--we seek to understand WHY theorems are true. (In contrast, lower division math courses often ask you to apply theorems that you are expected to take on faith.) It is an advantage to have had or to be taking Math 109, which bridges the gap between lower and upper division courses.

Exams Bring bluebook.   Calculators are not allowed. No make-up tests or early tests will be given, but scores will be prorated in the case of excused absences. Test 1:   Friday, April 25, 2008 Test 2:   Friday, May 23, 2008 Final:   (Take home final to be handed out on Friday, June 6, 2008) Solutions to Midterm #1 Solutions to Midterm #2 Solutions to Final Exam

Grading Homework:  25% Test 1:          25% Test 2:          25% Final Exam:   25%

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