Difference between revisions of "Number theory"

Revision as of 12:46, 30 July 2006

Number theory is the field of mathematics associated with studying the integers.

Introductory
- the Art of Problem Solving Introduction to Number Theory by Mathew Crawford (details)
General Interest
- Fermat's Enigma by Simon Singh (details)
- Music of the Primes by Marcus du Sautoy (details)

@@ Line 6: / Line 6: @@
 * [[Number theory/Intermediate | Intermediate topics in number theory]]
 * [[Number theory/Olympiad | Olympiad number topics in number theory]]
+* [[Number theory/Advanced topics | Advanced topics in number theory]]
-== Advanced Topics in Number Theory ==
-=== Algebraic Number Theory ===
-[[Algebraic number theory]] studies number theory from the perspective of [[abstract algebra]]. In particular, heavy use is made of [[ring theory]] and [[Galois theory]]. Algebraic methods are particularly well-suited to studying properties of individual prime numbers. From an algebraic perspective, number theory can perhaps best be described as the study of <math>\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})</math>. Famous problems in algebraic number theory include the [[Birch and Swinnerson-Dyer Conjecture]] and [[Fermat's Last Theorem]].
-=== Analytic Number Theory ===
-[[Analytic number theory]] studies number theory from the perspective of [[calculus]], and in particular [[real analysis]] and [[complex analysis]]. The techniques of [[analysis]] and [[calculus]] are particularly well-suited to studying large-scale properties of prime numbers. The most famous problem in analytic number theory is the [[Riemann Hypothesis]].
-=== Elliptic Curves and Modular Forms ===
-(I don't really feel like writing this right now. Any volunteers?)