Editing Theory of equations (section)

==History==
Until the end of the 19th century, "theory of equations" was almost synonymous with "algebra". For a long time, the main problem was to find the solutions of a single non-linear polynomial equation in a single [[Equation|unknown]]. The fact that a [[complex number|complex]] solution always exists is the [[fundamental theorem of algebra]], which was proved only at the beginning of the 19th century and does not have a purely algebraic proof. Nevertheless, the main concern of the algebraists was to solve [[Solution in radicals|in terms of radicals]], that is to express the solutions by a formula which is built with the four operations of [[arithmetics]] and with [[nth root]]s. This was done up to degree four during the 16th century. [[Scipione del Ferro]] and [[Niccolò Fontana Tartaglia]] discovered solutions for [[cubic equation]]s. [[Gerolamo Cardano]] published them in his 1545 book ''[[Ars Magna (Gerolamo Cardano)|Ars Magna]]'', together with a solution for the [[quartic equation]]s, discovered by his student [[Lodovico Ferrari]]. In 1572 [[Rafael Bombelli]] published his ''L'Algebra'' in which he showed how to deal with the [[imaginary number|imaginary quantities]] that could appear in Cardano's formula for solving cubic equations.

The case of higher degrees remained open until the 19th century, when [[Paolo Ruffini]] gave an incomplete proof in 1799 that some fifth degree equations cannot be solved in radicals followed by [[Niels Henrik Abel]]'s complete proof in 1824 (now known as the [[Abel–Ruffini theorem]]). [[Évariste Galois]] later introduced a theory (presently called [[Galois theory]]) to decide which equations are solvable by radicals.