Abstract
The equation y3 + hy2 ? + d = O may be transformed into the standard form x3 + px + q = 0 by the transformation y = x (h/3). The transformed equation was solved by Niccolo Tartaglia in the 16th century [1] by the method described below. If either equation has a rational root, the other will also. However, in contrast to the situation with quadratic equations, the computations carried out in the case of a rational root involve certain irrational, perhaps even complex, quantities. Historically, this represents the first serious involvement European mathematicians had with complex numbers. From a number theoretic point of view, it is interesting to investigate which irrational quantities appear in the application of Tartaglia's formula to equations with rational p and q and one or more rational roots. Although the process usually involves quantities from two distinct quadratic fields, we shall show exactly which special cases involve one quadratic field or just rational numbers. Solving the cubic X3 + pX + q by Tartaglia's method we have
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