p=x^2+54y^2 형태의 소수에 관한 연구

Abstract

In this thesis, we generate ring class fields of imaginary quadratic fields in terms of the special values of certain eta-quotients, which are related to the relative norms of Siegel-Ramachandra invariants. These give us minimal polynomials with relatively small coefficients from which we are able to solve the Diophantine equations $p=x^2+ny^2$. In particular, we classify all the primes of the form $x^2+54y^2$.