We investigate if
a given monic polynomial of the 3rd degree without quadratic term
P(X) = X^{3} + C*X + D with rational integer coefficients C,D
has a rational linear split factor
(X - r) with r in Z,
i. e., we test P(X) for reducibility.

In the case of irreducibility, we determine additionally
the polynomial discriminant d(P),
an integral basis of the shape (1, xi + a, xi^{2} + b*xi + c),
that is a unitary and in the zero xi of P(X) canonical
Z-basis of the maximal order O(L) of the cubic number field
L = Q(xi) generated by xi,
the polynomial index i(P) and the discriminant d(L) of L.
Here, we have the index relations
d(P) = i(P)^{2}*d(L)
and i(P) = (O(L):O), where O denotes the suborder of L with power Z-basis
(1, xi, xi^{2}).

The algorithm has been developed by VORONOI [1].

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