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Concept
Casus irreducibilis
Summary
In algebra, casus irreducibilis (Latin for "the irreducible case") is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed to numerically), i.e., by obtaining 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 have three real roots, which was proven by Pierre Wantzel in 1843.
One can see whether a given cubic polynomial is in so-called casus irreducibilis by looking at the discriminant, via Cardano's formula.
Let
be a cubic equation with . Then the discriminant is given by
It appears in the algebraic solution and is the square of the product
of the differences of the 3 roots .
If D < 0, then the polynomial has one real root and two complex non-real roots. is purely imaginary.Although there are cubic polynomials with negative discriminant which are irreducible in the modern sense, casus irreducibilis does not apply.
If D = 0, then and there are three real roots; two of them are equal. Whether D = 0 can be found out by the Euclidean algorithm, and if so, the roots by the quadratic formula. Moreover, all roots are real and expressible by real radicals.All the cubic polynomials with zero discriminant are reducible.
If D > 0, then is non-zero and real, and there are three distinct real roots which are sums of two complex conjugates.Because they require complex numbers (in the understanding of the time: cube roots from non-real numbers, i.e. from square roots from negative numbers) to express them in radicals, this case in the 16th century has been termed casus irreducibilis.
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 express a solution of p(x) = 0 by radicals with radicands ∈ F.
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