Resolvent cubic


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:
Suppose that the coefficients of belong to a field whose characteristic is different from . In other words, we are working in a field in which. Whenever roots of are mentioned, they belong to some extension of such that factors into linear factors in. If is the field of rational numbers, then can be the field of complex numbers or the field of algebraic numbers.
In some cases, the concept of resolvent cubic is defined only when is a quartic in depressed form—that is, when.
Note that the fourth and fifth definitions below also make sense and that the relationship between these resolvent cubics and are still valid if the characteristic of is equal to .

First definition

Suppose that is a depressed quartic—that is, that. A possible definition of the resolvent cubic of is:
The origin of this definition lies in applying Ferrari's method to find the roots of. To be more precise:
Add a new unknown,, to. 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 = 0.
If is a root of, then it is a consequence of the computations made [|above] that the roots of are the roots of the polynomial
together with the roots of the polynomial
Of course, this makes no sense if, but since the constant term of is, is a root of if and only if, and in this case the roots of can be found using the quadratic formula.

Second definition

Another possible definition is
The origin of this definition is similar to the previous one. This time, we start by doing:
and a computation similar to the previous one shows that this last expression is a square if and only if
A simple computation shows that

Third definition

Another possible definition is
The origin of this definition lies in another method of solving quartic equations, namely Descartes' method. If you try to find the roots of by expressing it as a product of two monic quadratic polynomials and, then
If there is a solution of this system with , the previous system is equivalent to
It is a consequence of the first two equations that then
and
After replacing, in the third equation, and by these values one gets that
and this is equivalent to the assertion that is a root of. So, again, knowing the roots of helps to determine the roots of.
Note that

Fourth definition

Still another possible definition is
In fact, if the roots of are, and, then
a fact the follows from Vieta's formulas. In other words, R4 is the monic polynomial whose roots are
,
, and
It is easy to see that
and
Therefore, has a multiple root if and only if has a multiple root. More precisely, and have the same discriminant.
One should note that if is a depressed polynomial, then

Fifth definition

Yet another definition is
If, as above, the roots of are, and, then
again as a consequence of Vieta's formulas. In other words, is the monic polynomial whose roots are
, and
It is easy to see that
and
Therefore, as it happens with, has a multiple root if and only if has a multiple root. More precisely, and have the same discriminant. This is also a consequence of the fact that = .
Note that if is a depressed polynomial, then

Applications

Solving quartic equations

It was explained above how Resolvent cubic#First definition|, Resolvent cubic#Second definition|, and Resolvent cubic#Third definition| can be used to find the roots of if this polynomial is depressed. In the general case, one simply has to find the roots of the depressed polynomial. For each root of this polynomial, is a root of .

Factoring quartic polynomials

If a quartic polynomial is reducible in, then it is the product of two quadratic polynomials or the product of a linear polynomial by a cubic polynomial. This second possibility occurs if and only if has a root in . In order to determine whether or not can be expressed as the product of two quadratic polynomials, let us assume, for simplicity, that is a depressed polynomial. Then it was seen above that if the resolvent cubic has a non-null root of the form, for some, then such a decomposition exists.
This can be used to prove that, in, every quartic polynomial without real roots can be expressed as the product of two quadratic polynomials. Let be such a polynomial. We can assume without loss of generality that is monic. We can also assume without loss of generality that it is a reduced polynomial, because can be expressed as the product of two quadratic polynomials if and only if can and this polynomial is a reduced one. Then = . There are two cases:
More generally, if is a real closed field, then every quartic polynomial without roots in can be expressed as the product of two quadratic polynomials in. Indeed, this statement can be expressed in first-order logic and any such statement that holds for also holds for any real closed field.
A similar approach can be used to get an algorithm to determine whether or not a quartic polynomial is reducible and, if it is, how to express it as a product of polynomials of smaller degree. Again, we will suppose that is monic and depressed. Then is reducible if and only if at least one of the following conditions holds:
Indeed:
The resolvent cubic of an irreducible quartic polynomial can be used to determine its Galois group ; that is, the Galois group of the splitting field of. Let be the degree over of the splitting field of the resolvent cubic. Then the group is a subgroup of the symmetric group. More precisely: