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In algebra, a septic equation is an equation of the form : where ''a'' ≠ 0. A septic function is a function of the form : where ''a'' ≠ 0. In other words, it is a polynomial of degree seven. If ''a'' = 0, then it is a sextic function (''b'' ≠ 0), quintic function (''b'' = 0, ''c'' ≠ 0), etc. The equation may be obtained from the function by setting ''y''(''x'') = 0. The ''coefficients'' may be either integers, rational numbers, real numbers, complex numbers or, more generally, members of any field. Because they have an odd degree, septic functions appear similar to quintic or cubic function when graphed, except they may possess additional local maxima and local minima (up to three maxima and three minima). The derivative of a septic function is a sextic function. ==Solvable septics== Some seventh degree equations can be solved by factorizing into radicals, but other septics cannot. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of Galois theory. To give an example of an irreducible but solvable septic, one can generalize the solvable de Moivre quintic to get, :, where the auxiliary equation is :. This means that the septic is obtained by eliminating ''u'' and ''v'' between , and . It follows that the septic's seven roots are given by : where ''ωk'' is any of the seven 7th roots of unity. The Galois group of this septic is the maximal solvable group of order 42. This is easily generalized to any other degrees ''k'', not necessarily prime. Another solvable family is, : whose members appear in Kluner's "Database of Number Fields". Its discriminant is, : Note that ''d'' = −467 has class number h(d) = 7. The Galois group of these septics is the dihedral group of order 14. The general septic equation can be solved with the alternating or symmetric Galois groups ''A''7 or ''S''7.〔 Such equations require hyperelliptic functions and associated theta functions of genus 3 for their solution.〔 However, these equations were not studied specifically by the nineteenth-century mathematicians studying the solutions of algebraic equations, because the sextic equations' solutions were already at the limits of their computational abilities without computers. Septics are the lowest order equations for which it is not obvious that their solutions may be obtained by superimposing continuous functions of two variables. Hilbert's 13th problem was the conjecture this was not possible in the general case for seventh-degree equations. Vladimir Arnold solved this in 1957, demonstrating that this was always possible. However, Arnold himself considered the ''genuine'' Hilbert problem to be whether the solutions of septics may be obtained by superimposing ''algebraic'' functions of two variables (the problem still being open). 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「Septic equation」の詳細全文を読む スポンサード リンク
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