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Topic: Quintic equation

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	EQUATION (from Lat. ae... - Online Information article about EQUATION (from Lat. ae...
		Thus we have the biquadratic equation y4+2Py22—84 R.y+P2-4Q=o, one of the roots of which is y= J a+ J b+ A) c, while a, b, c are the roots of the cubic equation z3+Pz2+Qz—R=o.
		Attempting to apply it to a quintic, we seek for the equation of which the root is (a+wb+w2c+wad+w4e), w an imaginary fifth root of unity, or rather the fifth power thereof (a+wb+w2c+wad+w4e)6; this is a 24-valued function, but if we consider the four values corresponding to the roots of unity w, w2, co3, w4, viz.
		This is, of course, useless for the solution of the quintic equation, which, as already mentioned, does not admit of solution by radicals; but the equation of the sixth order, Lagrange's re-solvent sextic, is very important, and is intimately connected with all the later investigations in the theory.
	encyclopedia.jrank.org /EMS_EUD/EQUATION_from_Lat_aequatio_aequ.html (11138 words)

	Britain.tv Wikipedia - Equation
		Equations are often used to state the equality of two expressions containing one or more variables.
		The two equations above are examples of identities: equations that are true regardless of the values of any variables that appear within them.
		However, if the equation were based on the natural numbers for example, some of these operations (like division and subtraction) may not be valid as negative numbers and non-whole numbers are not allowed.
	www.britain.tv /wikipedia.php?title=Equation (547 words)

	Introduction.
		Abel's proof of the insolvability of the general quintic polynomial appeared in 1826 [1]; later Galois gave the exact criterion for an equation to be solvable by radicals: its Galois group must be solvable.
		The quintic equation and the icosahedron are of course discussed at length in Klein's treatise [8] (see also [10], [2], [5], and especially [15]).
		The iterative scheme we use to solve the quintic relies on the map of degree 11 associated to the 12 vertices of the icosahedron.
	math.dartmouth.edu /~doyle/docs/icos/icos/node1.html (698 words)

	Quintic equation - Wikipedia, the free encyclopedia
		The derivative of a quintic function is a quartic function.
		Solving linear, quadratic, cubic and quartic equations by factorization into radicals is fairly straightforward when the roots are rational and real; 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.
	en.wikipedia.org /wiki/Quintic_equation (820 words)

	Encyclopedia :: encyclopedia : Quintic equation (Site not responding. Last check: 2007-11-02)
		In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five.
		The roots of the original equation are now expressible in terms of the roots of the transformed equation.
		We have a reduction to the Bring-Jerrard form in terms of solvable polynomial equations, and we used transformations involving polynomial expressions in the roots only up to the fourth degree, which means inverting the transformation may be done by finding the roots of a polynomial solvable in radicals.
	www.hallencyclopedia.com /topic/Quintic_equation.html (864 words)

	Search Results for equation*
		He solved his operator equation in the particular cases which arise in the study of the physical problem in his thesis (and in the paper which appeared in 1900 based on that thesis) while the general case was solved by Fredholm somewhat later and not published until 1908.
		In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals as the title shows: General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible.
		However, at the time of Bring's discovery, there was no hint that the quintic could not be solved by radicals and, although Bring does not claim that he discovered his transformation in an attempt to solve the quintic, it is likely that this is in fact why he was examining quintic equations.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=equation*&CONTEXT=1 (17342 words)

	Solving the Quintic
		Finally, Ruffini (1799) and Abel (1826) showed that the solution of the general quintic cannot be written as a finite formula involving only the four arithmetic operations and the extraction of roots.
		The group structure of the icosahedral equation is related to the stereographic projection of the triangulated regular icosahedron.
		For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions.
	library.wolfram.com /examples/quintic/main.html (890 words)

	Visualizing solutions to n-th degree algebraic equations using right-angle geometric paths, based on Lill's Method. (Site not responding. Last check: 2007-11-02)
		In forming the right-angle path for a quadratic equation, the directions for the a, b, and c lines are individually absolute in the sense that a negative coefficient would be represented by a line in the opposite direction, without affecting the directions for the other coefficients.
		For a cubic equation, once one solution is found, it is easy to know whether or not the remaining two are real or complex by constructing the solution circle and seeing if it intersects with the b-line of the subpath quadratic.
		Note that the solutions, as the negative tangents of the angles between the sub-paths and the equation path, are opposite in sign from those of the original equation and are fractions with 6 as the denominator.
	www.concentric.net /~pvb/ALG/rightpaths.html (3794 words)

	Quintic equation (Site not responding. Last check: 2007-11-02)
		Finding the zeroes of a polynomial — values of x which satisfy such an equation — given its coefficients was long a prominent mathematical problem.
		But if there was some pattern to the formulæ none could see it, and the quintic was proving to be extremely stubborn.
		This is somewhat surprising; even though the zeroes exist, there is no single finite expression of +, -, ×, ÷, and radicals that can produce them from the coefficients for all quintics.
	publicliterature.org /en/wikipedia/q/qu/quintic_equation.html (258 words)

	Solutions to Polynomial Equations
		In the case of the cubic equation below, we will need to extract cube roots, so our notation will be more like that of (2).
		is significantly more difficult to solve than the quadratic equation, and its general solution was not found until the sixteenth century.
		The cubic formula is more complicated than the quadratic formula and cannot reasonably be written without a change of variables.
	www.math.rutgers.edu /~erowland/polynomialequations.html (363 words)

	Quintic Gallery (Site not responding. Last check: 2007-11-02)
		This new equation can then be transformed by the Canonical Transform into the form of x^5-x-p==0 which is the Hermite quintic equation.
		This is the Principle form of the quintic equation and is solved using Klein's method.
		Here is a quintic equation that I solved using some properties of the sine function.
	www.csh.rit.edu /~topher/math/quinticgallery.html (259 words)

	The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry (Site not responding. Last check: 2007-11-02)
		The story of the equation that couldn't be solved is a story of brilliant mathematicians and a fascinating account of how mathematics illuminates a wide variety of disciplines.
		The specific equation that couldn't be solved is the quintic, which cannot be factored in general.
		Mario Livio's title suggests an exploration of unsolvable equations, in particular the drama enshrouding the mathematical conundrum of solving general, fifth degree polynomial equations, known as quintics.
	www.cuppalove.com /Shopping/Details/0743258207.aspx (1575 words)

	Quintic Equation -- from Wolfram MathWorld
		Irreducible quintic equations can be associated with a Galois group, which may be a symmetric group
		In the case of a solvable quintic, the roots can be found using the formulas of Malfatti (1771), who was the first to "solve" the quintic using a resolvent of sixth degree (Pierpont 1895).
		By solving a quartic, a quintic can be algebraically reduced to the Bring quintic form, as was first done by Jerrard.
	mathworld.wolfram.com /QuinticEquation.html (689 words)

	MATLAB (Site not responding. Last check: 2007-11-02)
		There may be several different values of x, called roots, that satisfy a univariate polynomial equation (the values of x where the polynomial equals 0).
		A univariate polynomial equation of degree 1 (n = 1) constitutes a linear equation.
		when n = 3, it is a cubic equation; when n = 4, it is a quartic equation; when n = 5, it is a quintic equation.
	kahuna.sdsu.edu /~lowdermi/Lecture_02-05-03_files/slide0151.htm (145 words)

	C:\internet\id\webpage_standrews\webpage_uhi\mathematica3\quintic\differentiallnk1.html (Site not responding. Last check: 2007-11-02)
		We differentiate the quintic equation eqn with respect to rho:
		This is a system of linear equations for the coefficients aj.
		This is a general solution to the differential equation that depends on the four parameters C[1], C[2],, C[3], C[4].
	easyweb.easynet.co.uk /~plindsay/webpage_uhi/mathematica3/quintic/differentiallnk1.html (227 words)

	[No title]
		In other words, he showed you can't solve this equation by means of some souped-up version of the quadratic formula that just involves taking the coefficients a,b,c,d,e,f and adding, subtracting, multiplying, dividing and taking nth roots.
		In general, a quintic equation has 5 solutions - and there's no "best one", so your formula has got to be a formula for all five.
		Roughly speaking, he showed that the general quintic equation is completely symmetrical under permuting all 5 solutions, and that this symmetry group - the group of permutations of 5 things - can't be built up from the symmetry groups that arise when you take nth roots.
	math.ucr.edu /home/baez/twf_ascii/week201 (3487 words)

	Hamilton's Investigations into the Solvability of Polynomial Equations
		Hamilton published the following papers on the solvability of polynomial equations, and in particular on the question as to whether or not the general quintic polynomial is or is not solvable by radicals:
		Following the solution of general cubic and biquadratic (quartic) equations by Italian mathematicians in the sixteenth century, mathematicians had sought to find solve polynomial equations of the fifth and higher degrees, but without success.
		Galois had also made a penetrating study of the solvability of polynomial equations, but his work was only brought to the attention of the mathematical community by Joseph Liouville in the 1840s.
	www.maths.tcd.ie /pub/HistMath/People/Hamilton/Quintic.html (586 words)

	The Hindu : Magazine / People : Norwegian mathematical genius
		Abel was the first to establish rigorously that there is no general formula to solve all polynomial equations of degree five; subsequently Galois established the impossibility of the solvability of polynomials of degree at least five, and this led to the creation of group theory.
		He was not interested in proving that certain equations did not have a formula for solutions, or did not have solutions of a certain type.
		In the course on his investigations on the quintics, Abel was led to discuss the convergence of the binomial series.
	www.hindu.com /mag/2004/12/26/stories/2004122600610400.htm (3501 words)

	Hermite, Charles (1822-1901) -- from Eric Weisstein's World of Scientific Biography
		French mathematician who did brilliant work in many branches of mathematics, but was plagued by poor performance in exams as a student.
		However, on his own, he mastered Lagrange's memoir on the solution of numerical equations and Gauss's Disquisitiones Arithmeticae.
		This equation was later found to arise in the quantum mechanical treatment of the simple harmonic oscillator.
	scienceworld.wolfram.com /biography/Hermite.html (200 words)

	Ramanujan's Solvable Modular Equations
		Indeed both Hermite and Kronecker showed, in the middle of the last century, that the solution of a general quintic may be effected in terms of the solution of the 5th-order modular equation (5.12) and the roots may thus be given in terms of the theta functions.
		Modular equations can be computed fairly easily from (5.11) and even more easily in the associated variables u and v.
		It is, therefore, one thing to solve these equations, it is entirely another matter to present them with the economy of Ramanujan.
	www.cecm.sfu.ca /organics/papers/borwein/paper/html/node13.html (567 words)

	Inquiry into the Validity of A Method recently proposed by George B. Jerrard, Esq. (Site not responding. Last check: 2007-11-02)
		In particular, he claimed that his methods enabled one to solve quintic equations without in the process needing to solve any polynomial equation of degree greater than four.
		Hamilton found that Jerrard had indeed constructed a general method for transforming polynomial equations to simpler forms by means of suitable Tschirnhaus transformations, but that the transformations only yielded a non-trivial result if the degree of the original polynomial equation was sufficiently large.
		(Indeed Abel had shown that it was not possible to solve the general quintic equation by radicals, and Hamilton was later to publish a detailed exposition of Abel's proof in the Transactions of the Royal Irish Academy.
	www.maths.tcd.ie /pub/HistMath/People/Hamilton/Jerrard (316 words)

	Read This: The Equation that Couldn't Be Solved
		After discussing cubic equations (and quartic equations, which make a surprise cameo appearance in the above tale), the natural question to consider is the quintic equation, and how to find a solution to a general fifth degree polynomial equation.
		The title of Livio's book gives away the punchline which is almost certainly familiar to any mathematician — that there is no way to solve a general quintic equation and that many such equations have no easily expressible solution — but it is a very interesting tale of how this punchline was reached by different mathematicians.
		I also think that The Equation that Could Not Be Solved would be an excellent book for a student to pick up to get drawn into the world of abstract algebra, and I have already been recommending the book to friends in the other science departments on campus.
	www.maa.org /reviews/LivioEquation.html (1378 words)

	The Brioschi Quintic And Other One-Parameter Forms
		To transform the principal form to Brioschi form, given r,s,t the objective is to equate coefficients to have a system of three equations in three unknowns a,b,c.
		degree equation can also be reduced to the Valentiner sextic form (after Siegfried Valentiner(?)), which can be solved in terms of generalized hypergeometric functions, in a manner analogous to Klein’s solution of the general quintic by reducing it to the Brioschi form.
		Weisstein, E., “Quintic Equation”, CRC Concise Encyclopedia of Mathematics, Chapman and Hall/CRC, 1999, or at http://mathworld.wolfram.com.
	www.geocities.com /titus_piezas/Brioschi.html (1906 words)

	Wikinfo \| Quadratic equation (Site not responding. Last check: 2007-11-02)
		A quadratic equation with real or complex coefficients has two complex roots (i.e., solutions), although the two roots may be equal.
		See also: linear equation, cubic equation, quartic equation, quintic equation, fundamental theorem of algebra
		Images, some of which are used under the doctrine of Fair use or used with permission, may not be available.
	www.wikinfo.org /wiki.php?title=Quadratic_equation (148 words)

	The inverse of a quintic function
		Finding the inverse of a quadratic or a cubic equation is a lot easier, but with this quintc I am really lost.
		I found in a calculus book that I can use the Newton method to find the zeros of the inverse of this equation, but this method is really hard.
		I would like to know if there is a easier method to find the zeros of the inverse of this quintic equation.
	mathcentral.uregina.ca /QQ/database/QQ.09.03/chana1.html (378 words)

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