Study of ideal class groups of algebraic number fields by using Diophantine equations and generic polynomials

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In this research, we investigated Diophantine equations and generic polynomials in order to construct algebraic number fields whose ideal class groups are not cyclic. As a result, for a prime number p which is congruent to 3 modulo 4, we gave explicitly an infinite family of imaginary cyclic fields of degree p-1 whose ideal class groups have p-rank at least 2. We also gave a refinement of Spiegelung relation for p=3, and got a new infinite family of imaginary quadratic fields whose ideal class groups have 3-rank at least 2.