In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four:
P ( x ) = x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 . {\displaystyle P(x)=x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.}
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.
Definitions Suppose that the coefficients of P(x) belong to a field k whose characteristic is different from 2. 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 Q 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.
First definition Suppose that P(x) is a depressed quartic—that is, that a3 = 0. A possible definition of the resolvent cubic of P(x) is:
R 1 ( y ) = 8 y 3 + 8 a 2 y 2 + ( 2 a 2 2 − 8 a 0 ) y − a 1 2 . {\displaystyle R_{1}(y)=8y^{3}+8a_{2}y^{2}+(2{a_{2}}^{2}-8a_{0})y-{a_{1}}^{2}.}
The origin of this definition lies in applying Ferrari's method to find the roots of P(x). To be more precise:
P ( x ) = 0 ⟺ x 4 + a 2 x 2 = − a 1 x − a 0 ⟺ ( x 2 + a 2 2 ) 2 = − a 1 x − a 0 + a 2 2 4 . {\displaystyle {\begin{aligned}P(x)=0&\Longleftrightarrow x^{4}+a_{2}x^{2}=-a_{1}x-a_{0}\\&\Longleftrightarrow \left(x^{2}+{\frac {a_{2}}{2}}\right)^{2}=-a_{1}x-a_{0}+{\frac {{a_{2}}^{2}}{4}}.\end{aligned}}}
Add a new unknown, y, to x2 + a2/2. Now you have:
… excerpt ends here. Continue reading the full article.
