Number theory/Advanced topics
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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 
. 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
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