Counting number fields with finite Abelian Galois group of bounded conductor that can be described as the sum of two squares.

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The Hermite-Minkowski theorem states that for any positive constant B, there are only finitely many number fields of discriminant less than B. A natural question which then arises is whether we can count these number fields. In 1989 Wright proved this was possible for Abelian extensions, and other specific cases where this is possible were proved by Davenport, Heilbronn and Bhargava. The aim of this project is to prove that one can count the number fields with finite abelian Galois group whose conductor satisfies certain restrictions, namely that it is bounded and the sum of two squares.In order to do so I will define a counting function and study the associated Dirichlet series. In order to do this I will first apply class field theory, in order to be able to study the problem over the adeles, and then use harmonic analysis to study the Dirichlet series, now over the adeles. I will then use a Poisson summation formula to write the Dirichlet series in terms of the Fourier transforms as they are easier to explicitly calculate, and I can use the analytic properties of the Fourier transforms to verify the the analytic properties of the original Dirichlet series. Finally, I will use the Fourier transforms to derive the required asymptotic formula.Although I will be counting by conductor, a possible extension of this would be to count by discriminant instead.The texts I am using are the papers "Number fields with prescribed norms" By Loughran, Frei and Newton, the paper of Landau and that of Serre on finding the asymptotic of numbers representable by the sum of two squares and the book "Advanced analytic number theory: L functions" by Moreno as a reference on harmonic analysis.