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	~~{{Other uses\|Septic (disambiguation){{!}}Septic}}~~		The writer's name is Ernest Swider. Debt collecting is how I support his dad. What he loves doing is to camp but he's been taking on new things lately. For a while he's been in Iowa. She's been working with her website for some time now. Investigate for yourself here: https://www.sbo69.club/⚽[https://www.sbo69.club/⚽Sbobet.php Sbobet].php
	~~[[Image:Septic graph.svg\|thumb\|right\|233px\|Graph of a polynomial of degree 7, with 6 [[critical point (mathematics)\|critical points]].]]~~

	~~In [[mathematics]], a '''septic equation''~~' is ~~[[equation]] of the form~~

	~~:<math>ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\,</math>~~

	~~where a ≠ 0~~.

	~~A '''septic function'''~~ is ~~a [[Function (mathematics)\|function]] of the form~~

	~~:<math>y(x)=ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h\,</math>~~

	~~where ''a'' ≠ 0~~. ~~In other words, it~~ is ~~a [[polynomial]] of [[Degree of a polynomial\|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'' {{nowrap\|1={{math\|''a'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g'', ''h''}}}} may be either [[integers]], [[rational number]]s, [[real number]]s, [[complex number]]s or, more generally, members of any [[field (mathematics)\|field]].

	Because they have an odd degree, '''septic functions''' appear similar to [[quintic function\|quintic]] or [[cubic function]] when graphed, except they may possess additional [[Maxima and minima\|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,
	~~:<math>x^7+7ax^5+14a^2x^3+7a^3x+b = 0\,</math>,~~

	~~where the auxiliary equation is~~
	~~:<math>y^2+by-a^7 = 0\,</math>.~~

	~~This means that the septic is obtained by eliminating ''u'' and ''v'' between~~
	~~<math>x=u+v </math>, <math>uv+a=0 </math> and <math>u^7+v^7+b=0 </math>.~~

	~~It follows that that the septic~~'s ~~seven roots are given by~~

	~~:<math>x_k = \omega_k\sqrt[7]{y_1} + \omega_k^6\sqrt[7]{y_2}</math>~~

	where ''ω<sub>k</sub>'' is any of the seven 7th [[root of unity\|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,~~

	~~:<math>x^7-2x^6+(~~a~~+1)x^5+(a-1)x^4-ax^3-(a+5)x^2-6x-4 = 0\,</math>~~

	~~whose members appear in Kluner~~'s ~~"Database of Number Fields"~~. ~~Its discriminant is,~~

	~~:<math>d = -4^4(4a^3+99a^2-34a+467)^3\,</math>~~

	~~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 group\|alternating]] or [[symmetric group\|symmetric]] [[Galois group]]s ''A''<sub>7</sub> or ''S''<sub>7</sub>.<ref name="BeyondQuartic"/> Such equations require [[hyperelliptic function]]s and associated [[theta function]]s of [[genus (mathematics)\|genus]] 3 for ~~their solution~~.<ref name="BeyondQuartic"/> However, these equations were not studied specifically by the nineteenth-century mathematicians studying the solutions of algebraic equations, because the [[sextic equation]]s' solutions were already at the limits of their computational abilities without computers.<ref name="BeyondQuartic">{{citation\|url=http://books.google.co.uk/books?id=9cKX_9zkeg4C&pg=PA143&lpg=PA143&dq=septic+equation&source=bl&ots=nld9eMx3DE&sig=wZ9V5zL0vNqsJvCguye-NCzqhq0&hl=en&ei=aF4oS570JdGHkQWd-936DA&sa=X&oi=book_result&ct=result&resnum=7&ved=0CDMQ6AEwBg#v=onepage&q=septic%20equation&f=false \|author=R. Bruce King \|title=Beyond the Quartic Equation \|publisher= Birkhaüser\|page= 143 and 144}}</ref>

	~~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 thirteenth problem\|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.<ref>{{citation \|url=http://books.google.co.uk/books?id=SpTv44Ia-J0C&pg=PA254 \|title=Kolmogorov's heritage in mathematics \|author=Vasco Brattka \|chapter=Kolmogorov's Superposition Theorem\|publisher=Springer}}</ref> 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).<ref>{{citation \|url=http://www.~~pdmi.ras.ru/~arnsem/Arnold/arnlect1.ps.gz \|title=From Hilbert's Superposition Problem to Dynamical Systems \|author=V.I. Arnold \|page=4}}</ref>~~

	~~==Galois groups==~~
	~~[[Image:Fano plane.svg\|thumb\|[[Fano plane]]]]~~
	*Septic equations solvable by radicals have a [[Galois group]] which is either the [[cyclic group]] of order 7, or the [[dihedral group]] of order 14 or a [[metacyclic group]] of order 21 or 42.<ref name="BeyondQuartic"/>
	*The ''L''(3, 2) [[Galois group]] (of order 168) is formed by the [[permutations]] of the 7 vertex labels which preserve the 7 "lines" in the [[Fano plane]].~~<ref name="BeyondQuartic"~~/~~> Septic equations with this~~ [~~[Galois group]] ''L''(3, 2) require [[elliptic function]]s but not [[hyperelliptic function]]s for their solution.<ref name="BeyondQuartic"/>~~
	*Otherwise the Galois group of a septic is either the [[alternating group]] of order 2520 or the [[symmetric group]] of order 5040.

	~~==Septic equation for the squared area of a cyclic pentagon or hexagon==~~

	The square of the area of a [[Pentagon#Cyclic pentagons\|cyclic pentagon]] is a root of a septic equation whose coefficients are [[symmetric function]]s of the sides of the pentagon.<ref>Weisstein, Eric W. "Cyclic Pentagon." From MathWorld--A Wolfram Web Resource. [http://~~mathworld~~.~~wolfram~~.~~com~~/~~CyclicPentagon~~.~~html~~]</ref> The same is true of the square of the area of a [[Hexagon#Cyclic hexagon\|cyclic hexagon]].<ref>Weisstein, Eric W. "Cyclic Hexagon." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/CyclicHexagon.~~html]</ref>~~

	~~==See also==~~
	*[[Cubic function]]
	*[[Quartic function]]
	*[[Quintic function]]
	*[[Sextic equation]]

	~~==References==~~
	~~<references/>~~

	~~{{Polynomials}}~~

	~~{{DEFAULTSORT:Septic Equation}}~~
	~~[[Category:Equations]]~~
	~~[[Category:Galois theory]]~~
	~~[[Category:Polynomials]]~~

165.112.184.36: /* Keto enol equilibria */

2013-04-09T21:37:16Z

Keto enol equilibria

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