Concept
In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: In each case: The coefficients of the resolvent cubic can be obtained from the coefficients of P(x) using only sums, subtractions and multiplications. Knowing the roots of the resolvent cubic of P(x) is useful for finding the roots of P(x) itself. Hence the name “resolvent cubic”. The polynomial P(x) has a multiple root if and only if its resolvent cubic has a multiple root. Suppose that the coefficients of P(x) belong to a field k whose characteristic is different from 2. In other words, we are working in a field in which 1 + 1 ≠ 0. Whenever roots of P(x) are mentioned, they belong to some extension K of k such that P(x) factors into linear factors in K[x]. If k is the field Q of rational numbers, then K can be the field C of complex numbers or the field of algebraic numbers. In some cases, the concept of resolvent cubic is defined only when P(x) is a quartic in depressed form—that is, when a3 = 0. Note that the fourth and fifth definitions below also make sense and that the relationship between these resolvent cubics and P(x) are still valid if the characteristic of k is equal to 2. Suppose that P(x) is a depressed quartic—that is, that a3 = 0. A possible definition of the resolvent cubic of P(x) is: The origin of this definition lies in applying Ferrari's method to find the roots of P(x). To be more precise: Add a new unknown, y, to x2 + a2/2. Now you have: If this expression is a square, it can only be the square of But the equality is equivalent to and this is the same thing as the assertion that R1(y) = 0. If y0 is a root of R1(y), then it is a consequence of the computations made above that the roots of P(x) are the roots of the polynomial together with the roots of the polynomial Of course, this makes no sense if y0 = 0, but since the constant term of R1(y) is –a12, 0 is a root of R1(y) if and only if a1 = 0, and in this case the roots of P(x) can be found using the quadratic formula.
