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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]].]]
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| In [[mathematics]], a '''septic equation''' is [[equation]] of the form
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| :<math>ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\,</math>
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| where a ≠ 0.
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| A '''septic function''' is a [[Function (mathematics)|function]] of the form
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| :<math>y(x)=ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h\,</math>
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| 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.
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| The equation may be obtained from the function by setting ''y''(''x'') = 0.
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| 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]].
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| 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]].
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| ==Solvable septics==
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| 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,
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| :<math>x^7+7ax^5+14a^2x^3+7a^3x+b = 0\,</math>,
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| where the auxiliary equation is
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| :<math>y^2+by-a^7 = 0\,</math>.
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| This means that the septic is obtained by eliminating ''u'' and ''v'' between
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| <math>x=u+v </math>, <math>uv+a=0 </math> and <math>u^7+v^7+b=0 </math>.
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| It follows that that the septic's seven roots are given by
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| :<math>x_k = \omega_k\sqrt[7]{y_1} + \omega_k^6\sqrt[7]{y_2}</math>
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| 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.
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| Another solvable family is,
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| :<math>x^7-2x^6+(a+1)x^5+(a-1)x^4-ax^3-(a+5)x^2-6x-4 = 0\,</math>
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| whose members appear in Kluner's "Database of Number Fields". Its discriminant is,
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| :<math>d = -4^4(4a^3+99a^2-34a+467)^3\,</math>
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| Note that ''d'' = −467 has [[class number]] h(d) = 7. The [[Galois group]] of these septics is the [[dihedral group]] of order 14.
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| 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>
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| 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>
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| ==Galois groups==
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| [[Image:Fano plane.svg|thumb|[[Fano plane]]]]
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| *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"/>
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| *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"/>
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| *Otherwise the Galois group of a septic is either the [[alternating group]] of order 2520 or the [[symmetric group]] of order 5040.
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| ==Septic equation for the squared area of a cyclic pentagon or hexagon==
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| 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>
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| ==See also==
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| *[[Cubic function]]
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| *[[Quartic function]]
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| *[[Quintic function]]
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| *[[Sextic equation]]
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| ==References==
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| <references/>
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| {{Polynomials}}
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| {{DEFAULTSORT:Septic Equation}}
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| [[Category:Equations]]
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| [[Category:Galois theory]]
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| [[Category:Polynomials]]
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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/⚽Sbobet.php