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| [[File:Polynomialdeg5.svg|thumb|right|233px|Graph of a polynomial of degree 5, with 4 [[critical point (mathematics)|critical points]]]]
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| In [[mathematics]], a '''quintic function''' is a [[function (mathematics)|function]] of the form
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| :<math>g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,</math>
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| where ''a'', ''b'', ''c'', ''d'', ''e'' and ''f'' are members of a [[field (mathematics)|field]], typically the [[rational number]]s, the [[real number]]s or the [[complex number]]s, and ''a'' is nonzero. In other words, a quintic function is defined by a [[polynomial]] of [[Degree of a polynomial|degree]] five.
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| If ''a'' is zero but one of the coefficients ''b'', ''c'', ''d'', or ''e'' is non-zero, the function is classified as either a [[quartic function]], [[cubic function]], [[quadratic function]] or [[linear function]].
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| Because they have an odd degree, normal quintic functions appear similar to normal [[cubic function]]s when graphed, except they may possess an additional [[Maxima and minima|local maximum]] and local minimum each. The [[derivative]] of a quintic function is a [[quartic function]].
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| Setting ''g''(''x'') = 0 and assuming ''a'' ≠ 0 produces a '''quintic [[equation]]''' of the form:
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| :<math>ax^5+bx^4+cx^3+dx^2+ex+f=0.\,</math>
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| ==Finding roots of a quintic equation==
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| Finding the roots of a given polynomial has been a prominent mathematical problem.
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| Solving [[Linear equation|linear]], [[Quadratic equation|quadratic]], [[Cubic equation|cubic]] and [[quartic equation]]s by [[factorization]] into [[Nth root|radical]]s is fairly straightforward, no matter whether the roots are rational or irrational, real or complex; there are also formulae that yield the required solutions. However, there is no formula for general quintic equations over the rationals in terms of radicals; this is known as the [[Abel–Ruffini theorem]], first published in 1824, which was one of the first applications of [[group theory]] in algebra. This result also holds for equations of higher degrees. An example quintic whose roots cannot be expressed by radicals is <math>x^5 - x + 1 = 0.</math> This quintic is in [[Bring–Jerrard normal form]].
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| As a practical matter, exact analytic solutions for polynomial equations are often unnecessary, and so numerical methods such as [[Laguerre's method]] or the [[Jenkins-Traub method]] are probably the best way of obtaining solutions to general quintics and higher degree polynomial equations that arise in practice. However, analytic solutions are sometimes useful for certain applications, and many mathematicians have tried to develop them.
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| ===Solvable quintics===
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| Some fifth-degree equations can be solved by factorizing into radicals; for example, <math>x^5 - x^4 - x + 1 = 0\,</math>, which can be written as <math>(x^2 + 1) (x + 1) (x - 1)^2 = 0\,</math>, or <math>x^5 -2 = 0\,</math>, which has <math>\sqrt[5]{2}\,</math> as solution. Other quintics like <math>x^5 - x + 1 = 0\,</math> cannot be solved by radicals. [[Évariste Galois]] developed techniques for determining whether a given equation could be solved by radicals which gave rise to [[group theory]] and [[Galois theory]]. Applying these techniques, [[Arthur Cayley]] found a general criterion for determining whether any given quintic is solvable.<ref>A. Cayley. ''On a new auxiliary equation in the theory of equation of the fifth
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| order'', Philosophical Transactions of the Royal Society of London (1861).</ref> This criterion is the following.<ref>This formulation of Cayley's result is extracted from Lazard (2004) paper.</ref>
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| Given the equation
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| :<math> ax^5+bx^4+cx^3+dx^2+ex+f=0,</math>
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| the [[Tschirnhaus transformation]] <math> x=y-\frac{b}{5a}</math>, which depresses the quintic, gives the equation
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| :<math> y^5+p y^3+q y^2+r y+s=0</math>,
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| where
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| : <math>\begin{align}
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| p &= \frac{5ac-2b^2}{5a^2}\\
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| q &= \frac{25a^2d-15abc+4b^3}{25a^3}\\
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| r &= \frac{125a^3e-50a^2bd+15ab^2c-3b^4}{125a^4}\\
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| s &= \frac{3125 a^4f-625a^3 be+125a^2b^2 d-25ab^3 c+4 b^5}{3125a^5}.
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| \end{align}
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| </math>
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| Both equations are solvable by radicals if and only if either they are factorisable in equations of lower degrees with rational coefficients or the polynomial <math>P^2-1024z\Delta</math>, named ''Cayley resolvent'', has a rational root in ''z'', where
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| :<math>
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| P =z^3-z^2(20r+3p^2)- z(8p^2r - 16pq^2- 240r^2 + 400sq - 3p^4)
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| </math>
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| ::<math>
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| {} - p^6 + 28p^4r- 16p^3q^2- 176p^2r^2- 80p^2sq + 224prq^2- 64q^4
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| </math>
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| ::<math>
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| {} + 4000ps^2 + 320r^3- 1600rsq
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| </math>
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| and
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| :<math>\Delta=-128p^2r^4+3125s^4-72p^4qrs+560p^2qr^2s+16p^4r^3+256r^5+108p^5s^2
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| </math>
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| ::<math>
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| {} -1600qr^3s+144pq^2r^3-900p^3rs^2+2000pr^2s^2-3750pqs^3+825p^2q^2s^2
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| </math>
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| ::<math>
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| {} +2250q^2rs^2+108q^5s-27q^4r^2-630pq^3rs+16p^3q^3s-4p^3q^2r^2.
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| </math>
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| In 1888, [[George Paxton Young]]<ref>George Paxton Young. ''Solvable Quintics Equations with Commensurable Coefficients'' ''American Journal of Mathematics'' '''10''' (1888), 99–130 [http://www.jstor.org/pss/2369502 at JSTOR]</ref> described how to solve a solvable quintic equation, without providing an explicit formula; [[Daniel Lazard]] wrote out a three-page formula (Lazard (2004)).
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| During the second half of 19th century, [[John Stuart Glashan]], George Paxton Young, and [[Carl Runge]] found that any [[irreducible polynomial|irreducible]] quintic with rational coefficients in [[Erland Samuel Bring|Bring]]-[[George Jerrard|Jerrard]] form,
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| :<math>x^5 + ax + b = 0\,</math>
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| is solvable by radicals if and only if either ''a'' = 0 or it is of the following form:
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| :<math>x^5 + \frac{5\mu^4(4\nu + 3)}{\nu^2 + 1}x + \frac{4\mu^5(2\nu + 1)(4\nu + 3)}{\nu^2 + 1} = 0</math>
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| where <math>\mu</math> and <math>\nu</math> are rational. In 1994, Blair Spearman and Kenneth S. Williams gave an alternative,
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| :<math>x^5 + \frac{5e^4( 4c + 3)}{c^2 + 1}x + \frac{-4e^5(2c-11)}{c^2 + 1} = 0.</math>
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| The relationship between the 1885 and 1994 parameterizations can be seen by defining the expression
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| :<math>b = \frac{4}{5} \left(a+20 \pm 2\sqrt{(20-a)(5+a)}\right)</math>
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| where
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| :<math>a = \frac{5(4\nu+3)}{\nu^2+1}</math>
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| and using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second. It is then a necessary (but not sufficient) condition that the irreducible solvable quintic
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| :<math>z^5 + a\mu^4z + b\mu^5 = 0\,</math>
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| with rational coefficients must satisfy the simple quadratic curve
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| :<math>y^2 = (20-a)(5+a)\,</math>
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| for some rational <math>a, y</math>.
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| The substitution <math>c=-m/l^5</math>, <math>e=1/l</math> in Spearman-Williams parameterization allows to not exclude the special case ''a'' = 0, giving the following result:
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| If ''a'' and ''b'' are rational numbers, the equation <math>x^5+ax+b=0</math> is solvable by radicals if either its left hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers ''l'' and ''m'' such that
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| :<math>a=\frac{5 l (3 l^5-4 m)}{m^2+l^{10}}\qquad b=\frac{4(11 l^5+2 m)}{m^2+l^{10}}</math>.
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| ===Examples of solvable quintics===
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| A quintic is solvable using radicals if the [[Galois group]] of the quintic (which is a subgroup of the [[symmetric group]] ''S''<sub>5</sub> of all permutations of a five element set) is a [[solvable group]]. In this case the form of the solutions depends on the structure of this Galois group.
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| A simple example is given by the equation <math>x^5-5x^4+30x^3-50x^2+55x-21=0\,,</math> whose Galois group is the group ''F''<sub>5</sub> generated by the cyclic permutations (1 2 3 4 5) and (1 2 4 3); the only real solution is
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| : <math>x=1+\sqrt[5]{2}-\sqrt[5]{4}+\sqrt[5]{8}-\sqrt[5]{16}\,.</math>
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| However, for other solvable Galois groups, the form of the roots can be much more complex. For example, the equation <math>x^5-5x+12=0\,</math> has Galois group ''D''<sub>5</sub> generated by "(1 2 3 4 5)" and "(1 4)(2 3)" and the solution requires more symbols to write. Define
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| :<math>a = \sqrt{2/\phi}</math>
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| :<math>b = \sqrt{2\phi}</math>
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| :<math>c = 5^{1/4}\,,</math>
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| where <math>\phi = (1+\sqrt{5})/2</math> is the [[golden ratio]], then the only real solution <math>x = -1.84208\dots</math> is given by
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| : <math>-(5^{1/4})x = \sqrt[5]{(a+c)^2(b-c)} + \sqrt[5]{(-a+c)(b-c)^2} + \sqrt[5]{(a+c)(b+c)^2} - \sqrt[5]{(-a+c)^2(b+c)} \,,</math>
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| or, equivalently, by
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| :<math>x = y_1^{1/5}+y_2^{1/5}+y_3^{1/5}+y_4^{1/5}\,,</math>
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| where the ''y<sub>i</sub>'' are the four roots of the [[quartic equation]]
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| :<math>y^4+4y^3+\frac{4}{5}y^2-\frac{8}{5^3}y-\frac{1}{5^5}=0\,.</math>
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| In general, if an equation P(x) = 0 of prime degree ''p'' with rational coefficients is solvable in radicals, then each root is the sum of ''p''-th roots of the roots of an auxiliary equation Q(y) = 0 of degree (''p''-1), also with rational coefficients, that can be used to solve the former. However these ''p''-th roots may not be computed independently (this would provide ''p''<sup>''p''</sup> roots instead of ''p''). Thus a correct solution needs to express all these ''p''-roots in term of one of them. Galois theory shows that this is always theoretically possible, even if the resulting formula may be too large to be of any use.
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| It is possible that some of the roots of Q(y) = 0 are rational (as in the above example with the ''F''<sub>5</sub> Galois group) or some are zero. When it is the case the formula for the roots is much simpler, like for the solvable [[de Moivre]] quintic{{anchor|de Moivre quintic}}
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| :<math>x^5+5ax^3+5a^2x+b = 0\,,</math>
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| where the auxiliary equation has two zero roots and reduces, by factoring them out, to the [[quadratic equation]]
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| :<math>y^2+by-a^5 = 0\,,</math>
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| such that the five roots of the de Moivre quintic are given by
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| :<math>x_k = \omega^k\sqrt[5]{y_i} -\frac{a}{\omega^k\sqrt[5]{y_i}},</math>
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| where ''y<sub>i</sub>'' is any root of the auxiliary quadratic equation and ''ω'' is any of the four [[primitive root of unity|primitive 5th roots of unity]]. This can be easily generalized to construct a solvable [[septic equation|septic]] and other odd degrees, not necessarily prime.
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| Here is a list of known solvable quintics:
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| There are infinitely many solvable quintics in Bring-Jerrard form which have been parameterized in preceding section.
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| Up to the scaling of the variable, there are exactly five solvable quintics of the shape <math>x^5+ax^2+b</math>, which are<ref>http://www.math.harvard.edu/~elkies/trinomial.html</ref> (where ''s'' is a scaling factor):
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| :<math>x^5-2s^3x^2-\frac{s^5}{5} </math>
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| :<math> x^5-100s^3x^2-1000s^5</math>
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| :<math>x^5-5s^3x^2-3s^5 </math>
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| :<math>x^5-5s^3x^2+15s^5 </math>
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| :<math> x^5-25s^3x^2-300s^5</math>
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| Paxton Young (1888) gave a number of examples, some of them being reducible, having a rational root:
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| {| <math>x^5+3x^2+2x-1 </math> ||
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| | <math> x^5-10x^3-20x^2-1505x-7412</math> ||
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| |-
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| | <math>x^5+\frac{625}{4}x+3750 </math>||
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| |-
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| | <math>x^5-\frac{22}{5}x^3-\frac{11}{25}x^2+\frac{462}{125}x+\frac{979}{3125} </math> ||
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| | <math>x^5+20x^3+20x^2+30x+10 </math> ||Solution: <math> \sqrt[5]{2}-\sqrt[5]{2}^2+\sqrt[5]{2}^3-\sqrt[5]{2}^4</math>
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| | <math> x^5+320x^2-1000x+4288</math> || Reducible: −8 is a root
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| |<math> x^5+40x^2-69x+108</math> || Reducible: −4 is a root
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| |-
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| |<math>x^5-20x^3+250x-400 </math> ||
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| |-
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| | <math>x^5-5x^3+\frac{85}{8}x-13/2 </math> ||
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| |-
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| |<math> x^5+\frac{20}{17}x+\frac{21}{17}</math> ||
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| |-
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| |<math>x^5-\frac{4}{13}x+\frac{29}{65}</math>||
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| |-
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| |<math> x^5+\frac{10}{13}x+\frac{3}{13} </math> ||
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| |-
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| | <math> x^5+110(5x^3+60x^2+800x+8320)</math> ||
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| | <math>x^5-20 x^3 -80 x^2 -150 x -656 </math>||
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| |-
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| | <math> x^5 -40 x^3 +160 x^2 +1000 x -5888 </math> ||
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| |-
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| |<math> x^5-50x^3-600x^2-2000x-11200</math> ||
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| |-
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| | <math>x^5+110(5 x^3 + 20x^2 -360 x +800) </math> ||
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| | <math> x^5- 20 x^3 +320 x^2 +540 x + 6368 </math> || Reducible : -8 is a root
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| |-
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| | <math>x^5-20 x^3 -160 x^2 -420 x -8928 </math>|| Reducible : 8 is a root
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| |-
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| | <math> x^5-20 x^3 +170 x + 208</math>||
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| |}
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| An infinite sequence of solvable quintics may be constructed, whose roots are sums of ''n''-th [[roots of unity]], with ''n'' = 10''k'' + 1 being a prime number:
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| {|
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| | <math>x^5+x^4-4x^3-3x^2+3x+1</math> || Roots: <math>2\cos(\frac{2k\pi}{11})</math>
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| |-
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| | <math> x^5+x^4-12x^3-21x^2+x+5</math>|| Root: <math> \sum_{k=0}^5 e^\frac{2i\pi 6^k }{31}</math>
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| | <math>y^5+y^4-16y^3+5y^2+21y-9 </math>|| Root: <math>\sum_{k=0}^7 e^\frac{2i\pi 3^k }{41}</math>
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| | <math>y^5+y^4-24y^3-17y^2+41y-13</math>|| Root: <math>\sum_{k=0}^{11} e^\frac{2i\pi (21)^k }{61}</math>
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| |-
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| | <math>y^5+y^4-28y^3+37y^2+25y+1</math>|| Root: <math>\sum_{k=0}^{13} e^\frac{2i\pi (23)^k }{71}</math>
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| |}
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| There are also two parameterized families of solvable quintics:
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| The Kondo–Brumer quintic,
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| :<math>x^5+(a-3)x^4+(-a+b+3)x^3+(a^2-a-1-2b)x^2+bx+a = 0\,</math>
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| and the family depending on the parameters <math>a, l, m</math>
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| :<math> x^5-5p(2x^3 + ax^2 + bx)-pc = 0\,</math>
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| where
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| ::<math>p=\frac{l^2(4m^2+a^2)-m^2}{4}, \qquad b=l(4m^2+a^2)-5p-2m^2, \qquad c=\frac{b(a+4m)-p(a-4m)-a^2m}{2}</math>
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| ===Beyond radicals===
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| {{main|Bring radical}}
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| If the Galois group of a quintic is not solvable, then the [[Abel-Ruffini theorem]] tells us that to obtain the roots it is necessary to go beyond the basic arithmetic operations and the extraction of radicals. About 1835, [[George Jerrard|Jerrard]] demonstrated that quintics can be solved by using [[ultraradical]]s (also known as [[Bring radical]]s), the real roots of <math>t^5 + t - a=0\,</math> for real numbers <math>a\,</math>. In 1858 [[Charles Hermite]] showed that the Bring radical could be characterized in terms of the Jacobi [[theta functions]] and their associated [[elliptic modular function]]s, using an approach similar to the more familiar approach of solving [[cubic equation]]s by means of [[trigonometric function]]s. At around the same time, [[Leopold Kronecker]], using [[group theory]] developed a simpler way of deriving Hermite's result, as had [[Francesco Brioschi]]. Later, [[Felix Klein]] came up with a method that relates the symmetries of the [[icosahedron]], [[Galois theory]], and the elliptic modular functions that feature in Hermite's solution, giving an explanation for why they should appear at all, and developed his own solution in terms of [[generalized hypergeometric function]]s.<ref>{{Harv|Klein|1888}}; a modern exposition is given in {{Harv|Tóth|2002|loc=Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, [http://books.google.com/books?id=i76mmyvDHYUC&pg=PA66 p. 66]}}</ref> Similar phenomena occur in degree 7 ([[septic equation]]s) and 11, as studied by Klein and discussed in [[Icosahedral symmetry#Related geometries|icosahedral symmetry: related geometries]].
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| ==See also==
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| *[[Cubic function]]
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| *[[Quartic function]]
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| *[[Sextic equation]]
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| *[[Septic function]]
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| *[[Theory of equations]]
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| *[[Solvable group]]
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| *[[Newton's method]]
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| * Charles Hermite, "Sur la résolution de l'équation du cinquème degré",''Œuvres de Charles Hermite'', t.2, pp. 5–21, Gauthier-Villars, 1908.
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| * Felix Klein, ''Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree'', trans. George Gavin Morrice, Trübner & Co., 1888. ISBN 0-486-49528-0.
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| * Leopold Kronecker, "Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite", ''Comptes Rendus de l'Académie des Sciences," t. XLVI, 1858 (1), pp. 1150–1152.
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| * Blair Spearman and Kenneth S. Williams, "Characterization of solvable quintics <math>x^5 + ax + b</math>", ''American Mathematical Monthly'', Vol. 101 (1994), pp. 986–992.
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| * Ian Stewart, ''Galois Theory'' 2nd Edition, Chapman and Hall, 1989. ISBN 0-412-34550-1. Discusses Galois Theory in general including a proof of insolvability of the general quintic.
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| * [[Jörg Bewersdorff]], ''Galois theory for beginners: A historical perspective'', American Mathematical Society, 2006. ISBN 0-8218-3817-2. Chapter 8 ([http://www.ams.org/bookstore/pspdf/stml-35-prev.pdf The solution of equations of the fifth degree]) gives a description of the solution of solvable quintics <math>x^5 + cx + d</math>.
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| * Victor S. Adamchik and David J. Jeffrey, "Polynomial transformations of Tschirnhaus, Bring and Jerrard," ''ACM SIGSAM Bulletin'', Vol. 37, No. 3, September 2003, pp. 90–94.
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| * Ehrenfried Walter von Tschirnhaus, "A method for removing all intermediate terms from a given equation," ''ACM SIGSAM Bulletin'', Vol. 37, No. 1, March 2003, pp. 1–3.
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| * Daniel Lazard, "Solving quintics in radicals", in [[Olav Arnfinn Laudal]], [[Ragni Piene]], ''The Legacy of Niels Henrik Abel'', pp. 207–225, Berlin, 2004,. ISBN 3-540-43826-2. available at http://www.loria.fr/publications/2002/A02-R-449/A02-R-449.ps (broken link)
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| * {{citation | title = Finite Möbius groups, minimal immersions of spheres, and moduli
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| | first = Gábor | last = Tóth | year = 2002 }}
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| {{refend}}
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| ==External links==
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| * [http://www.had2know.com/academics/quintic-equation-solver-calculator.php Quintic Equation Solver]
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| * [http://mathworld.wolfram.com/QuinticEquation.html Mathworld - Quintic Equation] – more details on methods for solving Quintics.
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| * [http://library.wolfram.com/examples/quintic/ Solving the Quintic with Mathematica] – poster on Quintic solutions
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| * [http://www.archive.org/details/cu31924059413439] – Klein's book is available online
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| * [http://www.emba.uvm.edu/~dummit/quintics/solvable.pdf Solving Solvable Quintics] – a method for solving solvable quintics due to David S. Dummit.
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| * [http://www.sigsam.org/bulletin/articles/145/Adamchik.pdf Polynomial Transformations of Tschirnhaus, Bring and Jerrard] - a recent update of Tschirnhaus' paper by Victor S. Adamchik & David J. Jeffrey
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| * [http://www.sigsam.org/bulletin/articles/143/tschirnhaus.pdf A method for removing all intermediate terms from a given equation] - a recent English translation of Tschirnhaus' 1683 paper.
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| {{Polynomials}}
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| {{DEFAULTSORT:Quintic Equation}}
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| [[Category:Equations]]
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| [[Category:Galois theory]]
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| [[Category:Polynomials]]
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