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A general solution to any quadratic equation can be given using the quadratic formula above. Similar formulas exist for polynomial equations of degree 3 and 4. But no such formula is possible for 5th degree polynomials; the real solution -1.1673... to the 5th degree equation below cannot be written using basic arithmetic operations and ''n''th roots: 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 with arbitrary coefficients.〔Jacobson (2009), p. 211.〕 The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799, and Niels Henrik Abel, who provided a proof in 1823. Évariste Galois independently proved the theorem in a work that was posthumously published in 1846.〔 〕 == Interpretation == The theorem does ''not'' assert that some higher-degree polynomial equations have ''no'' solution. In fact, the opposite is true: ''every'' non-constant polynomial equation in one unknown, with real or complex coefficients, has at least one complex number as a solution (and thus, by polynomial division, as many complex roots as its degree, counting repeated roots); this is the fundamental theorem of algebra. These solutions can be computed to any desired degree of accuracy using numerical methods such as the Newton–Raphson method or Laguerre method, and in this way they are no different from solutions to polynomial equations of the second, third, or fourth degrees. The theorem only shows that there is no ''general'' solution in radicals that applies to all equations of a given high degree. Also, the theorem does not preclude the possibility that every higher degree equation has its own idiosyncratic solution in radicals. That is, the theorem does not assert that some higher-degree polynomial equations have roots which cannot be expressed in terms of radicals. While this is now known to be true, it is a stronger claim, which was only proven a few years later by Galois. The theorem only shows that there is no ''general'' solution in terms of radicals which gives the roots to a generic polynomial with arbitrary coefficients. It did not by itself rule out the possibility that each polynomial may be solved in terms of radicals on a case-by-case basis. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Abel–Ruffini theorem」の詳細全文を読む スポンサード リンク
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