Description
In this project,we explore several topics in the algebraic
theory of arithmetic. Among these we will discuss
- Quadratic Reciprocity:
You will have seen this in year 2, but now we revisit
the result and give a more conceptual approach(es).
- p-adic numbers:
These `new' numbers are an essential tool in number
theory. Essentially, they give us a way to study
congruence to arbitrary high prime powers
simultaneously. However, this beautiful topic is of its
own interest.
- Quadratic Forms: These are homogeneous
quadratic polynomials in several variables. They are of
classic interest in number theory and we will study them
of the integers, the rationals, and the p-adic numbers.
In particular we will establish the Local-Global
principle, which allows us to derive properties over the
rationals from the properties over the p-adic numbers.
Resources
We will roughly follow the classic book
- J-P Serre: A Course in Arithmetic. Graduate Texts in
Mathematics 7, Springer.
Depending on
interest we will then also consult other resources.
Prerequisites
- Elementary Number Theory II
- Algebra II
email: J Funke
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