Number theory

Revision as of 14:52, 12 July 2006 by MCrawford (talk | contribs) (adding subpage links and exporting intro material)

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


Student Guides to Number Theory


Intermediate Topics

An intermediate level of study involves many of the topics of introductory number theory, but involves an infusion of mathematical problem solving as well as algebra.


Olympiad Topics

An Olympiad level of study involves familiarity with intermediate topics to a high level, a few new topics, and a highly developed proof writing ability.

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 $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. 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?)


Resources

Books

Miscellaneous


Other Topics of Interest

These are other topics that aren't particularly important for competitions and problem solving, but are good to know.


Famous Unsolved Number Theory Problems