When pure cubic fields are monogenic?
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Pure cubic fields are considered monogenic when the proportion of monogenic orders among the orders whose indices are a power of a fixed prime is analyzed. Research indicates that for prime numbers other than 2 or 3, this proportion is zero . Furthermore, in the context of square-free integers and finite fields, it has been shown that the number of square-free integers or polynomials leading to monogenic cubic fields exhibits specific growth patterns and bounds, highlighting the conditions under which these fields are monogenic . By employing various mathematical criteria and techniques, these studies shed light on the monogeneity of pure cubic fields under different mathematical contexts and conditions.
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09 Apr 2022 | Not addressed in the paper. |
23 Jun 2023 | Pure cubic fields are monogenic when a fixed prime number is 2 or 3. For other prime numbers, the proportion of monogenic orders among orders with prime power indices is zero. |
| Pure cubic fields are monogenic when specific conditions are met, such as square-free integers and certain field properties, leading to a significant number of monogenic instances. | |
| Pure cubic fields are monogenic when the number field L = K(α) satisfies a simple version of Dedekind’s criterion, allowing for efficient characterization of monogeneity over K. | |
| Pure cubic fields are monogenic when the proportion of monogenic orders among orders with prime power indices is zero for prime numbers other than 2 or 3. |
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