His solution contained a gap, which Cauchy considered minor, though this was not patched until the work of Norwegian mathematician Niels Henrik Abel, who published a proof in 1824, thus establishing the Abel – Ruffini theorem.

Among his work was an incomplete proof ( Abel – Ruffini theorem ) that quintic ( and higher-order ) equations cannot be solved by radicals ( 1799 ), and Ruffini's rule which is a quick method for polynomial division.

Ruffini ’ s 1799 work marked a major development for group theory.

The theory of field extensions ( including Galois theory ) involves the roots of polynomials with coefficients in a field ; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel – Ruffini theorem on the algebraic insolubility of quintic equations.

The quintic was almost proven to have no general solutions by radicals by Paolo Ruffini in 1799, whose key insight was to use permutation groups, not just a single permutation.

While Ruffini and Abel established that the general quintic could not be solved, some particular quintics can be solved, such as ( x − 1 )< sup > 5 </ sup >= 0, and the precise criterion by which a given quintic or higher polynomial could be determined to be solvable or not was given by Évariste Galois, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots – in modern terms, its Galois group – had a certain structure – in modern terms, whether or not it was a solvable group.

This is a result of Galois theory ( see Quintic equations and the Abel – Ruffini theorem ).

Although Abel had already proved the impossibility of a " quintic formula " by radicals in 1824 and Ruffini had published a solution in 1799 that turned out to be flawed, Galois ' methods led to deeper research in what is now called Galois theory.

The birth of Galois theory was originally motivated by the following question, whose answer is known as the Abel – Ruffini theorem.

Historically, Ruffini and Abel's proofs precede Galois theory.

One of the fundamental theorems of Galois theory states that an equation is solvable in radicals if and only if it has a solvable Galois group, so the proof of the Abel – Ruffini theorem comes down to computing the Galois group of the general polynomial of the fifth degree.

However, there is no formula for general quintic equations over the rationals in terms of radicals ; this is known as the Abel – Ruffini theorem, first published in 1824, which was one of the first applications of group theory in algebra.

In an 1850 paper, On the Solution of the General Equation of the Fifth Degree, Kronecker solved the quintic equation by applying group theory ( though his solution was not in terms of radicals, since this was already proven impossible by Abel – Ruffini theorem ).

Ruffini also made contributions to group theory in addition to probability and quadrature of the circle.

These methods are named after the British mathematician William George Horner, although they were known before him by Paolo Ruffini and, six hundred years earlier, by the Chinese mathematician Qin Jiushao.

Don Pasquale is an opera buffa, or comic opera, in three acts by Gaetano Donizetti with an Italian libretto by Giovanni Ruffini and the composer after Angelo Anelli's libretto for Stefano Pavesi's Ser Marc ' Antonio ( 1810 ).

The primary sensory receptors for touch / position ( Meissner ’ s corpuscles, Merkel's receptors, Pacinian corpuscles, Ruffini ’ s corpuscles, hair receptors, muscle spindle organs, and Golgi tendon organs ) are structurally more complex than the primitive receptors for pain / temperature, which are bare nerve endings.

* Ruffini ’ s end organ detects temperatures below body temperature

la: Paulus Ruffini

In algebra, the Abel – Ruffini theorem ( also known as Abel's impossibility theorem ) states that there is no general algebraic solution — that is, solution in radicals — to polynomial equations of degree five or higher.

The Abel – Ruffini theorem says that there are some fifth-degree equations whose solution cannot be so expressed.

Niels Henrik Abel ( 1802 – 1829 ), a Norwegian, and Évariste Galois, ( 1811 – 1832 ) a Frenchman, investigated into the solutions of various polynomial equations, and proved that there is no general algebraic solution to equations of degree greater than four ( Abel – Ruffini theorem ).

Ruffini was the first to controversially assert the unsolvability by radicals of algebraic equations higher than quartics.

The latter is also known as Ruffini – Horner's method.

For, in 1823, Abel had at last proved the impossibility of solving the quintic equation in radicals ( now referred to as the Abel – Ruffini theorem ).

Higher-degree polynomials have no such general solution, according to the Abel – Ruffini theorem ( 1824, 1799 ).

The roots of a cubic, like those of a quadratic or quartic ( fourth degree ) function but no higher degree function ( by the Abel – Ruffini theorem ), can always be found algebraically ( as a formula involving simple functions like the square root and cube root functions ).

The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the Abel – Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile.

* Angelo Ruffini, ( 1864 – 1929 ), Italian histologist and embryologist

* Ernesto Ruffini ( 1888 – 1967 ), Archbishop of Palermo

* Giovanni Ruffini ( 1807 – 1881 ), Italian poet and librettist

* Paolo Ruffini ( 1765 – 1822 ), Italian mathematician and philosopher

* Eugen Ruffínyi ( 1846 – 1924 ), Slovak mining engineer and amateur speleologist ; original family name was pribably " Ruffini "

Ruffini assumed that a solution would necessarily be a function of the radicals ( in modern terms, he failed to prove that the splitting field is one of the fields in the tower of radicals which corresponds to a solution expressed in radicals ).

Ruffini endings function as static mechanoreceptors which position the mandible.

He is remembered through the so-called Ruffini house at Munich's Rindermarkt, which he bought in 1708, and through his tombstone at the outer wall of St. Peter's Church in Munich.

Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots which is a special case of the methods given many centuries later by Ruffini and Horner.

The repression was ruthless: 12 participants were executed, while Mazzini's best friend and director of the Genoese section of the Giovine Italia, Jacopo Ruffini, killed himself.

The theorem, however, was first nearly proved by Paolo Ruffini in 1799, but his proof was mostly ignored.

Thorne ( 1994 ) relates that this approach to studying black holes was prompted by the realisation by Hanni, Ruffini, Wald and Cohen in the early 1970s that since an electrically charged pellet dropped into a black hole should still appear to a distant outsider to be remaining just outside the event horizon, if its image persists, its electrical fieldlines ought to persist too, and ought to point to the location of the " frozen " image ( 1994, pp. 406 ).

The SM. 79 flown by Sottotenente Ruffini was hit by anti-aircraft fire from a British warship and crashed.

Paolo Ruffini ( September 22, 1765 – May 10, 1822 ) was an Italian mathematician and philosopher.

In mathematics, Ruffini's rule is an efficient technique for dividing a polynomial by a binomial of the form x − r. It was described by Paolo Ruffini in 1809.

Kleinert's 60. birthday was honored by a Festschrift and a Festcolloquium with 65 contributions by international colleagues ( for instance Y. Ne ' eman, R. Jackiw, H. Fritzsch, R. Ruffini, C. DeWitt, L. Kauffman, J. Devreese, and K. Maki ).