(A) Study of Class Number of Real Quadratic and Cubic Fields

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The primary goal of the thesis is to study class number of the ring of integers of a number field and related arithmetical properties. The first part of the thesis provides a solution to a classical problem posed by Dirichlet. Dirichlet asked whether exist infinitely many real quadratic fields with 1 as relative class number. It has been shown in the thesis that a real quadratic field will always have 1 as relative class number for the conductor 3, depending om certain conditions on the field . The thesis also provides a necessary and sufficient condition for a real quadratic field to have relative class number 1. Moreover, the thesis contains significant generalization of the continued fraction approach of A. Furness and E. A. Parker towards relative class number. Using this approach, the thesis also shows the existence of another infinite family of real quadratic fields with relative class number 1 with an odd prime dividing the discriminant as conductor.
Supervisor: Anupam Saikia