Definition of Principal Factorization Types (PFT):
The pure cubic number fields L = Q(R^{1/3}) can be classified into 3 principal factorization types according to the ambiguous principal ideals in their normal field N.
Among the generators of primitive (not divisible by an integer > 1) ambiguous principal ideals of a pure cubic field we can always find 2 radicals R^{1/3} and S^{1/3}, whose conjugates differ only by 3rd roots of unity. Here, the cubefree radicands R = mn^{2} and S = m^{2}n can be represented by squarefree coprime positive integers m > n.
Concerning further principal factors, we distinguish 3 cases:
Statistical Results:
In 1989 I have constructed an extensive database [1] for the 82264 pure cubic fields L = Q(R^{1/3}) with radicands in the range R < 100000 thereby finding the following distribution of the 3 principal factorization types:
Bibliography:
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