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  2. Abel–Ruffini theorem - Wikipedia

    en.wikipedia.org/wiki/AbelRuffini_theorem

    The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799 [1] (which was refined and completed in 1813 [2] and accepted by Cauchy) and Niels Henrik Abel, who provided a proof in 1824. [3] [4] AbelRuffini theorem refers also to the slightly stronger result that there are equations of degree five and higher that cannot be ...

  3. List of long mathematical proofs - Wikipedia

    en.wikipedia.org/wiki/List_of_long_mathematical...

    As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof. 1799 The AbelRuffini theorem was nearly proved by Paolo Ruffini, but his proof, spanning 500 pages, was mostly ignored and later, in 1824, Niels Henrik Abel published a proof that required just six pages.

  4. Ruffini's rule - Wikipedia

    en.wikipedia.org/wiki/Ruffini's_rule

    Ruffini's rule can be used when one needs the quotient of a polynomial P by a binomial of the form . (When one needs only the remainder, the polynomial remainder theorem provides a simpler method.) A typical example, where one needs the quotient, is the factorization of a polynomial p ( x ) {\displaystyle p(x)} for which one knows a root r :

  5. Galois theory - Wikipedia

    en.wikipedia.org/wiki/Galois_theory

    The AbelRuffini theorem provides a counterexample proving that there are polynomial equations for which such a formula cannot exist. Galois' theory provides a much more complete answer to this question, by explaining why it is possible to solve some equations, including all those of degree four or lower, in the above manner, and why it is ...

  6. Eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Eigenvalue_algorithm

    The AbelRuffini theorem shows that any such algorithm for dimensions greater than 4 must either be infinite, or involve functions of greater complexity than elementary arithmetic operations and fractional powers. For this reason algorithms that exactly calculate eigenvalues in a finite number of steps only exist for a few special classes of ...

  7. Niels Henrik Abel - Wikipedia

    en.wikipedia.org/wiki/Niels_Henrik_Abel

    Abel sent a paper on the unsolvability of the quintic equation to Carl Friedrich Gauss, who proceeded to discard without a glance what he believed to be the worthless work of a crank. [12] As a 16-year-old, Abel gave a rigorous proof of the binomial theorem valid for all numbers, extending Euler's result which had held only for rationals.

  8. List of theorems - Wikipedia

    en.wikipedia.org/wiki/List_of_theorems

    ATS theorem (number theory) Abel's binomial theorem (combinatorics) Abel's curve theorem (mathematical analysis) Abel's theorem (mathematical analysis) Abelian and Tauberian theorems (mathematical analysis) Abel–Jacobi theorem (algebraic geometry) AbelRuffini theorem (theory of equations, Galois theory) Abhyankar–Moh theorem (algebraic ...

  9. Quartic equation - Wikipedia

    en.wikipedia.org/wiki/Quartic_equation

    The proof that this was the highest order general polynomial for which such solutions could be found was first given in the AbelRuffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile.