Casus irreducibilis (from Latin 'the irreducible case') is the name given by mathematicians of the 16th century to cubic equations that cannot be solved in terms of real radicals, that is to those equations such that the computation of the solutions cannot be reduced to the computation of square and cube roots.
Cardano's formula for solution in radicals of a cubic equation was discovered at this time.
It applies in the casus irreducibilis, but, in this case, requires the computation of the square root of a negative number, which involves knowledge of complex numbers, unknown at the time.
The casus irreducibilis occurs when the three solutions are real and distinct, or, equivalently, when the discriminant is positive.
It is only in 1843 that Pierre Wantzel proved that there cannot exist any solution in real radicals in the casus irreducibilis.
Then the discriminant is given by It appears in the algebraic solution and is the square of the product of the
Then casus irreducibilis states that it is impossible to express a solution of p(x) = 0 by radicals with real radicands.
Since this is F or a quadratic extension of F (depending in whether or not D is a square in F), p(x) remains irreducible in it.
Then p(x) can be split by a tower of cyclic extensions At the final step of the tower, p(x) is irreducible in the penultimate field K, but splits in K(3√α) for some α.
But this is a cyclic field extension, and so must contain a conjugate of 3√α and therefore a primitive 3rd root of unity.
However, there are no primitive 3rd roots of unity in a real closed field.
Then, by the axioms defining an ordered field, ω and ω2 are both positive, because otherwise their cube (=1) would be negative.
The equation ax3 + bx2 + cx + d = 0 can be depressed to a monic trinomial by dividing by
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 q2/4 + p3/27 < 0, 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 α + βi with specifically given expressions in real radicals for α and β, 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.
Since its discriminant is positive, it has three real roots, so it is an example of casus irreducibilis.
The solutions are in radicals and involve the cube roots of complex conjugate numbers.
While casus irreducibilis cannot be solved in radicals in terms of real quantities, it can be solved trigonometrically in terms of real quantities.
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.
On the other hand, if the rational root test shows that there is no rational root, then casus irreducibilis applies, cos(θ⁄3) or sin(θ⁄3) is not constructible, the angle θ⁄3 is not constructible, and the angle θ is not classically trisectible.
Expressing cos(20°) in radicals results in which involves taking the cube root of complex numbers.
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.
Casus irreducibilis can be generalized to higher degree polynomials as follows.
Then the degree of p is a power of 2, and its splitting field is an iterated quadratic extension of F.[7][8]: 571–572 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, i.e. it is a casus irreducibilis in the (16th century) sense of this article.
Casus irreducibilis for quintic polynomials is discussed by Dummit.
[9]: 17 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 θ, one-fifth of this angle has a cosine that is one of the five real roots of the equation Likewise, θ/5 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—x1) 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, cos(θ⁄5) and sin(θ⁄5) are not constructible, the angle θ⁄5 is not constructible, and the angle θ is not classically pentasectible.
While a pentagon is relatively easy to construct, a 25-gon requires an angle pentasector as the minimal polynomial for cos(14.4°) has degree 10: Thus, It may be noticed that
occurs in Cardano’s formula (as well as the primitive 3rd roots of unity