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Fermat proved that a rational prime p is the sum of two squares if and only if p=4k+1. Similarly, one can prove without too much trouble that a rational prime p is of the form x^2+2y^2 if and only if p=8k+1 or p=8k+3.
Although basic algebraic number theory suffices to classify primes of the form X^2+y^2 and x^2+2y^2, the more general problem of classifying primes of the form x^2+ny^2 (where n is a square-free integer) is best done with class field theory.
During this talk we'll provide the class field theoretic background needed to classify primes of the form x^2+ny^2. The focus will be on examples and generalizing some of the concepts mentioned last week, during our discussion of class field theory over the rationals.
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