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UBC Theses and Dissertations

Tau-congruent numbers and integers in number fields Davis, Chad Tyler


A positive integer n is the area of a Heron triangle if and only if there exists a non-zero rational number tau such that the elliptic curve E^n_tau: Y² = X(X-n tau)(X+n tau-¹) has a rational point of order different than two. Such an integer n is called a tau-congruent number. In the first part of this thesis, we give two distribution theorems on tau-congruent numbers; in particular we show that given any fixed, non-zero rational tau, there exist infinitely many tau-congruent numbers in every residue class modulo any positive integer m, and we also prove a similar result for tau-congruent numbers whose corresponding elliptic curves have rank at least 2. In the second half of this thesis, we give some theorems on integers in certain algebraic number fields. First, we completely determine the index of a quartic field defined by a polynomial of the form X⁴ + aX + b. Second, we determine all possible normal integral bases in a certain cyclic quintic field. Finally, we give a family of cubic polynomials that define number fields that are both monogenic and have the same discriminant.

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