Abstract

Motivated by a constructive realization of dihedral groups of prime degree as Galois group over the field of rational numbers, we give an explicit construction of the Hilbert class fields of some imaginary quadratic fields with class numbers 7 and 11. This was done by explicitly evaluating the elliptic modular j -invariant at each representative of the ideal class of an imaginary quadratic field, and then forming the class equation. In an appendix we determine the explicit form of the modular equation of order 5 and 7. All computations were carried out on the computer algebra system MACSYMA. We also point out what computational difficulties were encountered and how we resolved them. In this paper we summarize the progress made so far on using the Computer Algebra System MACSYMA [8] to explicitly calculate the defining equations of the Hilbert class fields of imaginary quadratic fields with prime class number. Our motivation for undertaking this investigation is to construct rational po...

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