Casus irreducibilis

In algebra, casus irreducibilis (Latin for "the irreducible case") is one of the cases that may arise in attempting to solve polynomials of degree 3 or higher with integer coefficients, to obtain roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of "casus irreducibilis" is in the case of cubic polynomials that are irreducible (cannot be factored into lower-degree polynomials) over the rational numbers and have three real roots, which was proven by Pierre Wantzel in 1843.
One can decide whether a given irreducible cubic polynomial is in "casus irreducibilis" using the discriminant "Δ", 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
Formal statement and proof.
More generally, suppose that "F" is a formally real field, and that "p"("x") ∈ "F"["x"] is a cubic polynomial, irreducible over "F", but having three real roots (roots in the real closure of "F"). Then "casus irreducibilis" states that it is impossible to find any solution of "p"("x") = 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 (depending in whether or not is a square in ), 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 formula_4 and substituting (the Tschirnhaus transformation), giving the equation where
Then regardless of the number of real roots, by Cardano's solution the three roots are given by
where formula_8 ("k"=1, 2, 3) is a cube root of 1 (formula_9, formula_10, and formula_11, where is the imaginary unit). 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 (the cube root cannot itself be placed in the form with specifically given expressions in real radicals for and , since doing so would require independently solving the original cubic). 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 (unnecessarily in this case) expresses that root (and the others) in terms of non-real radicals.
Example.
The cubic equation
is irreducible, because if it could be factored there would be a linear factor giving a rational solution, while none of the possible roots given by the rational root test are actually roots. Since its discriminant is positive, it has three real roots, so it is an example of "casus irreducibilis." These roots can be expressed as
for formula_14. The solutions are in radicals and involve the cube roots of complex conjugate numbers.
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 formula_15 is solved by
These solutions are in terms of real quantities if and only if formula_17 — 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 (arccosine) are transcendental functions, this solution is algebraic in the sense that formula_18 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 is trisectible by the classical means of compass and unmarked straightedge. For any angle , one-third of this angle has a cosine that is one of the three solutions to
Likewise, has a sine that is one of the three real solutions to
In either case, if the rational root test reveals a rational solution, 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.
As an example, while a 180° angle can be trisected into three 60° angles, a 60° angle cannot be trisected with only compass and straightedge. Using triple-angle formulae one can see that where . Rearranging gives , which fails the rational root test as none of the rational numbers suggested by the theorem is actually a root. Therefore, the minimal polynomial of has degree 3, whereas the degree of the minimal polynomial of any constructible number must be a power of two.
Expressing in radicals results in
which involves taking the cube root of complex numbers. Note the similarity to and .
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 (mechanically, by simply drawing a diagonal).
Generalization.
"Casus irreducibilis" can be generalized to higher degree polynomials as follows. Let "p" ∈ "F"["x"] be an irreducible polynomial which splits in a formally real extension "R" of "F" (i.e., "p" has only real roots). Assume that "p" has a root in formula_24 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 (quintisection) and higher.
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 (able to be split into five equal parts) by the classical means of compass and unmarked straightedge. For any angle , one-fifth of this angle has a cosine that is one of the five real roots of the equation
Likewise, 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 ("x—x"1) 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. An example of this is when one attempts to construct a 25-gon (icosipentagon) with compass and straightedge. While a pentagon is relatively easy to construct, a 25-gon requires an angle pentasector as the minimal polynomial for has degree 10:
Thus,