Back حالة غير قابلة للتبسيط (معادلات تكعيبية) Arabic Casus irreducibilis German Casus irreducibilis Spanish Casus irreducibilis French Casus irreducibilis Hungarian 환원 불능의 경우 Korean Casus irreducibilis Latin Casus irreducibilis Portuguese Casus irreducibilis Russian Casus irreducibilis Ukrainian
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.[1]
- ^ Wantzel, Pierre (1843), "Classification des nombres incommensurables d'origine algébrique" (PDF), Nouvelles Annales de Mathématiques (in French), 2: 117–127