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Minimal indices of pure cubic fields Yoo, Jeewon


Determining whether a number field admits a power integral basis is a classical problem in algebraic number theory. It is well known that every quadratic number field is monogenic, that is they admit power bases. However, when we talk about cubic or higher degree number fields we may discover fields without power integral bases. In 1878, Dedekind gave the first example of a cubic field without a power integral basis. It is known that a number field is monogenic if and only if the minimal index is one. In 1937, Hall proved that the minimal index of pure cubic fields can be arbitrarily large. We extend this result by showing that the minimal index of a family of infinitely many pure cubic fields have an element of index n but no element of index less than n for a positive integer n.

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