Interactive Determination of the Reducibility of a Cubic Polynomial

Java applet for Cubic Polynomials


We investigate if a given monic polynomial of the 3rd degree without quadratic term P(X) = X3 + 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, xi2 + 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, xi2).

The algorithm has been developed by VORONOI [1].


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