When a polynomial's solution cannot be expressed in radicals without complex numbers
In algebra, casus irreducibilis (from Latin '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.[1]
One can see whether a given cubic polynomial is in so-called casus irreducibilis by looking at the discriminant, via Cardano's formula.[2]
^Wantzel, Pierre (1843), "Classification des nombres incommensurables d'origine algébrique" (PDF), Nouvelles Annales de Mathématiques (in French), 2: 117–127
^Cox (2012), Theorem 1.3.1, p. 15.
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