Casus irreducibilis


In algebra, casus irreducibilis is one of the cases that may arise in attempting to solve polynomials of degree 3 with integer coefficients, to obtain roots that are expressed with radicals. Specifically, in the case of cubic polynomials, if a cubic is irreducible over the rational numbers and has three real roots, then expressing the roots with radicals requires one to introduce complex numbers, even though the resulting expressions are ultimately real-valued. This was proven by Pierre Wantzel in 1843.
One can decide whether a given irreducible cubic polynomial is in casus irreducibilis using the discriminant D, via Cardano's formula. Let the cubic equation be given by
with a≠0. Then the discriminant appearing in the algebraic solution is given by
More generally, suppose that F is a formally real field, and that p ∈ F is a cubic polynomial, irreducible over F, but having three real roots. Then casus irreducibilis states that it is impossible to find any solution of p = 0 by real radicals.
To prove this, note that the discriminant is positive. Form the field extension. Since this is or a quadratic extension of , remains irreducible in it. Consequently, the Galois group of over is the cyclic group. Suppose that can be solved by real radicals. Then can be split by a tower of cyclic extensions
At the final step of the tower, is irreducible in the penultimate field, but splits in for some. But this is a cyclic field extension, and so must contain a primitive root of unity.
However, there are no primitive 3rd roots of unity in a real closed field. Suppose that ω is a primitive 3rd root of unity. Then, by the axioms defining an ordered field, ω, ω2, and 1 are all positive. But if ω2>ω, then cubing both sides gives 1>1, a contradiction; similarly if ω>ω2.

Solution in non-real radicals

Cardano's solution

The equation can be depressed to a monic trinomial by dividing by and substituting , giving the equation where
Then regardless of the number of real roots, by Cardano's solution the three roots are given by
where is a cube root of 1. Here if the radicands under the cube roots are non-real, the cube roots expressed by radicals are defined to be any pair of complex conjugate cube roots, while if they are real these cube roots are defined to be the real cube roots.
Casus irreducibilis occurs when none of the roots are rational and when all three roots are distinct and real; the case of three distinct real roots occurs if and only if, in which case Cardano's formula involves first taking the square root of a negative number, which is imaginary, and then taking the cube root of a complex number. Even in the reducible case in which one of three real roots is rational and hence can be factored out by polynomial long division, Cardano's formula expresses that root in terms of non-real radicals.

Example

The depressed cubic equation
is irreducible, because if it could be factored there would be a linear factor giving a rational solution, while by the rational root test there is no rational root. Since its discriminant is positive, it has three real roots, so it is an example of casus irreducibilis. Cardano's formula gives these three real roots as
for. This solution in radicals involves the imaginary number and hence involves the cube roots of complex conjugate numbers.
As another example:
which is one of the solutions of

Trigonometric solution in terms of real quantities

While casus irreducibilis cannot be solved in radicals in terms of real quantities, it can be solved trigonometrically in terms of real quantities. Specifically, the depressed monic cubic equation is solved by
These solutions are in terms of real quantities if and only if — i.e., if and only if there are three real roots. The formula involves starting with an angle whose cosine is known, trisecting the angle by multiplying it by 1/3, and taking the cosine of the resulting angle and adjusting for scale.
Although cosine and its inverse function are transcendental functions, this solution is algebraic in the sense that is an algebraic function, equivalent to angle trisection.

Relation to angle trisection

The distinction between the reducible and irreducible cubic cases with three real roots is related to the issue of whether or not an angle with rational cosine or rational sine is trisectible by the classical means of compass and unmarked straightedge. If the cosine of an angle is known to have a particular rational value, then one third of this angle has a cosine that is one of the three real roots of the equation
Likewise, if the sine of is known to have a particular rational value, then one third of this angle has a sine that is one of the three real roots of the equation
In either case, if the rational root test reveals a rational root of the equation, or minus that root can be factored out of the polynomial on the left side, leaving a quadratic that can be solved for the remaining two roots in terms of a square root; then all of these roots are classically constructible since they are expressible in no higher than square roots, so in particular or is constructible and so is the associated angle. On the other hand, if the rational root test shows that there is no rational root, then casus irreducibilis applies, or is not constructible, the angle is not constructible, and the angle is not classically trisectible.
The connection between rational roots and trisectability can also be extended to some cases where the sine and cosine of the given angle is irrational. Consider as an example the case where the given angle is a vertex angle of a regular pentagon, a polygon that can be constructed classically. For this angle is 180°, and standard trigonometric identities then give
thus
The cosine of the trisected angle is rendered as a rational expression in terms of the cosine of the given angle, so the vertex angle of a regular pentagon can be trisected.

Generalization

Casus irreducibilis can be generalized to higher degree polynomials as follows. Let p ∈ F be an irreducible polynomial which splits in a formally real extension R of F. Assume that p has a root in which is an extension of F by radicals. Then the degree of p is a power of 2, and its splitting field is an iterated quadratic extension of F.
Thus for any irreducible polynomial whose degree is not a power of 2 and which has all roots real, no root can be expressed purely in terms of real radicals. Moreover, if the polynomial degree is a power of 2 and the roots are all real, then if there is a root that can be expressed in real radicals it can be expressed in terms of square roots and no higher-degree roots, as can the other roots, and so the roots are classically constructible.
Casus irreducibilis for quintic polynomials is discussed by Dummit.

Relation to angle pentasection

The distinction between the reducible and irreducible quintic cases with five real roots is related to the issue of whether or not an angle with rational cosine or rational sine is pentasectible by the classical means of compass and unmarked straightedge. If the cosine of an angle is known to have a particular rational value, then one fifth of this angle has a cosine that is one of the five real roots of the equation
Likewise, if the sine of is known to have a particular rational value, then one fifth of this angle has a sine that is one of the five real roots of the equation
In either case, if the rational root test yields a rational root x1, then the quintic is reducible since it can be written as a factor times a quartic polynomial. But if the test shows that there is no rational root, then the polynomial may be irreducible, in which case casus irreducibilis applies, and are not constructible, the angle is not constructible, and the angle is not classically pentasectible.
Pentasection of also implies pentasection of as:.