Principal Factors in Pure Cubic Fields

Definition of Principal Factorization Types (PFT):

The pure cubic number fields L = Q(R1/3) 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 R1/3 and S1/3, whose conjugates differ only by 3rd roots of unity. Here, the cubefree radicands R = mn2 and S = m2n 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(R1/3) with radicands in the range R < 100000 thereby finding the following distribution of the 3 principal factorization types:

For the most recent results see my series Joint Research 2002.


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