The proportion of monogenic orders of prime power indices of a pure cubic field
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In this paper, we investigate the proportion of monogenic orders among the orders whose indices are powers of a fixed prime in a pure cubic field. We prove that the proportion is zero for a prime number that is not equal to 2 or 3. To do this, we first count the number of orders whose indices are powers of a fixed prime. This is done by considering every full rank submodules of the ring of integers, and establishing the condition to be closed under multiplication. Next, we derive the index form of arbitrary orders based on the index form of the ring of integers. Then, we obtain an upper bound of the number of monogenic orders with prime power indices by applying the finiteness of the number of primitive solutions of a Thue-Mahler equation.
Abstract
In this paper, we investigate the proportion of monogenic orders among the orders whose indices are powers of a fixed prime in a pure cubic field. We prove that the proportion is zero for a prime number that is not equal to 2 or 3. To do this, we first count the number of orders whose indices are powers of a fixed prime. This is done by considering every full rank submodules of the ring of integers, and establishing the condition to be closed under multiplication. Next, we derive the index form of arbitrary orders based on the index form of the ring of integers. Then, we obtain an upper bound of the number of monogenic orders with prime power indices by applying the finiteness of the number of primitive solutions of a Thue-Mahler equation.
참고문헌 (7)
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