Fifth degree equation

A fifth degree equation or a quintic equation is a polynomial equation of degree five in mathematics , so it has the form

{\ displaystyle ax ^ {5} + bx ^ {4} + cx ^ {3} + dx ^ {2} + ex + f = 0,}

where the coefficients and elements of a body (typically the rational , real or complex numbers ) are with . One then speaks of an equation “about” this body. ${\ displaystyle a, b, c, d, e}$ ${\ displaystyle f}$ ${\ displaystyle a \ neq 0}$

Polynomial of degree 5:
f (x) = (x + 4) (x + 2) (x + 1) (x-1) (x-3) / 20 + 2

history

Solving polynomial equations by finite root expressions ( radicals ) is an old problem. After Gerolamo Cardano published solutions to the general equations up to degree 4 in his book Ars magna in 1545 , efforts focused on solving the general equation of the fifth degree. In 1771 Gianfrancesco Malfatti was the first to find a solution, which only works in the case of solvability through root expressions. In 1799, Paolo Ruffini published incomplete proof of the indissolubility of the general equation of the fifth degree. Since Ruffini used unfamiliar arguments for the time that are now assigned to group theory , his proof was initially not accepted. In 1824 Niels Henrik Abel succeeded in completely proving that the general equation of the fifth degree cannot be solved by radicals ( Abel-Ruffini theorem ). In Galois theory , the proof can be summarized as follows: The Galois group of the general equation -th degree has the alternating group as a factor, and this group is simple for (cf. icosahedral group ), i.e. not solvable. Charles Hermite succeeded in 1858 in solving the general fifth degree equation in Jacobian theta functions (but not in radicals, of course). ${\ displaystyle n}$ ${\ displaystyle A_ {n}}$ ${\ displaystyle n \ geq 5}$

Fifth degree solvable equations

Some fifth degree equations can be solved with roots, for example , which can be factored in the form . Other equations such as cannot be solved by roots. Évariste Galois developed methods around 1830 to determine whether a given equation is solvable in roots (see Galois theory ). Based on these basic results, George Paxton Young and Carl Runge proved in 1885 an explicit criterion for whether a given equation of the fifth degree with roots can be solved (cf. Lazard's work for a modern approach). They showed that an irreducible fifth degree equation with rational coefficients in Bring-Jerrard form ${\ displaystyle x ^ {5} -x ^ {4} -x + 1 = 0}$ ${\ displaystyle (x ^ {2} +1) (x + 1) (x-1) ^ {2} = 0}$ ${\ displaystyle x ^ {5} -x + 1 = 0}$

{\ displaystyle x ^ {5} + ax + b = 0}

is solvable with roots if and only if they take the form

{\ displaystyle 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}

with rational and owns. In 1994, Blair Spearman and Kenneth S. Williams found the representation ${\ displaystyle \ mu}$ ${\ displaystyle \ nu}$

{\ displaystyle x ^ {5} + {\ frac {5e ^ {4} (3-4c \ epsilon)} {c ^ {2} +1}} x + {\ frac {-4e ^ {5} (11 \ epsilon + 2c)} {c ^ {2} +1}} = 0}

for . The relationship between the two parameterizations can be given by the equation ${\ displaystyle \ epsilon = \ pm 1}$

{\ displaystyle b = (4/5) \ left (a + 20 + 2 {\ sqrt {(20-a) (5 + a)}} \ right)}

With

{\ displaystyle a = {\ frac {5 (4v + 3)} {v ^ {2} +1}}}

getting produced. In the case of the negative square root, the first parameterization is obtained with suitable scaling, and the second with a positive square root . Hence, it is a necessary (but not a sufficient) condition for a solvable equation of the fifth degree of form ${\ displaystyle \ epsilon = -1}$

{\ displaystyle z ^ {5} + a \ mu ^ {4} z + b \ mu ^ {5} = 0}

with rational , and that equation ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle \ mu}$

{\ displaystyle y ^ {2} = (20-a) (5 + a)}

has a rational solution . ${\ displaystyle y}$

With the help of Tschirnhaus transformations it is possible to bring every equation of the fifth degree into Bring-Jerrard form, therefore the parameterizations of Runge and Young as well as of Spearman and Williams give necessary and sufficient conditions to check whether any equation fifth degree is to be solved in radicals.

Examples of solvable equations of the fifth degree

An equation is solvable in radicals if its Galois group is a solvable group . For -th degree equations , their Galois group is a subgroup of the symmetric group , the permutations of elements. ${\ displaystyle n}$ ${\ displaystyle S_ {n}}$ ${\ displaystyle n}$

A simple example of a solvable equation is with the Galois group F (5) which is generated by the permutations “(1 2 3 4 5)” and “(1 2 4 3)”; is the only real root ${\ displaystyle x ^ {5} -5x ^ {4} -10x ^ {3} -10x ^ {2} -5x-1 = 0}$ ${\ displaystyle x = 1 + {\ sqrt [{5}] {2}} + {\ sqrt [{5}] {4}} + {\ sqrt [{5}] {8}} + {\ sqrt [ {5}] {16}}.}$

Euler gave the root for the equation . ${\ displaystyle x ^ {5} -40x ^ {3} -70x ^ {2} + 50x + 98 = 0}$ ${\ displaystyle x = {\ sqrt [{5}] {- 31 + 3 {\ sqrt {-7}}}} + {\ sqrt [{5}] {- 31-3 {\ sqrt {-7}} }} + {\ sqrt [{5}] {- 18 + 10 {\ sqrt {-7}}}} + {\ sqrt [{5}] {- 18-10 {\ sqrt {-7}}}} }$

However, the solutions can also be much more complex. For example, the equation has the Galois group D (5) which is generated by “(1 2 3 4 5)” and “(1 4) (2 3)”, and the solution takes about 600 symbols written out. ${\ displaystyle x ^ {5} -5x + 12 = 0}$

literature

Charles Hermite: Sur la résolution de l'équation du cinquième degré. In: Œuvres de Charles Hermite . Volume 2, pages 5-21, Gauthier-Villars, 1908 ( available online ).

Felix Klein : Lectures on the icosahedron and the solution of the equations of the fifth degree . Leipzig 1884, ISBN 0-486-49528-0 ( available online ).

Leopold Kronecker : Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite. In: Comptes Rendus de l'Académie des Sciences. Volume 66 No. 1, 1858, pages 1150-1152.

Blair Spearman and Kenneth S. Williams: Characterization of solvable quintics . ${\ displaystyle x ^ {5} + ax + b}$ In: American Mathematical Monthly . Volume 101, 1994, pages 986-992.

Bruce Berndt , Blair Spearman, Kenneth S. Williams, editors (Comments on an unpublished lecture of GN Watson On solving the quintic ) by GN Watson : On solving the quintic. Mathematical Intelligencer, Vol. 24, 2002, No.

Ian Stewart : Galois Theory. 2nd Edition. Chapman and Hall, 1989, ISBN 0-412-34550-1 .

Jörg Bewersdorff : Algebra for Beginners: From Equation Resolution to Galois Theory , Springer Spectrum, 5th Edition 2013, ISBN 978-3-658-02261-7 , doi : 10.1007 / 978-3-658-02262-4_8 . Chapter 8 describes the solution of solvable equations of the fifth degree in the form (book chapter in English translation The solution of equations of the fifth degree is available online ( Memento from March 31, 2010 in the Internet Archive ) (PDF file; 131 kB)). ${\ displaystyle x ^ {5} + cx + d = 0}$

Victor S. Adamchik and David J. Jeffrey: Polynomial transformations of Tschirnhaus, Bring and Jerrard. In: ACM SIGSAM Bulletin . Volume 37 No. 3, September 2003, pages 90-94 ( available online (PDF file; 140 kB)).

Ehrenfried Walther von Tschirnhaus : A method for removing all intermediate terms from a given equation. In: ACM SIGSAM Bulletin . Volume 37 No. 1, March 2003, pages 1-3.

Daniel Lazard: Solving quintics in radicals. In: Olav Arnfinn Laudal, Ragni Piene: The Legacy of Niels Henrik Abel . Berlin 2004, pages 207-225, ISBN 3-540-43826-2 .

Peter Pesic: Abel's proof . Springer 2005, ISBN 3-540-22285-5 .

Jean-Pierre Tignol: Galois' Theory of Algebraic Equations . World Scientific, 2004, ISBN 981-02-4541-6 , doi : 10.1142 / 9789812384904 .

DS Dummit Solving solvable quintics , Mathematics of Computation, Volume 57, 1991, pp. 387-402 (Corrigenda Volume 59, 1992, p. 309)

Individual evidence

^ Young, GP: Solution of Solvable Irreducible Quintic Equations, Without the Aid of a Resolvent Sextic. In: Amer. J. Math. Volume 7, pages 170-177, 1885.

↑ Runge, C .: About the solvable equations of the form . ${\ displaystyle x ^ {5} + ux + v = 0}$ In: Acta Math. Volume 7, pages 173-186, 1885, doi : 10.1007 / BF02402200 .

↑ George Jerrard found a method to eliminate the terms of the order ( n -1), ( n -2), ( n -3) in equations of the nth degree by a polynomial transformation , which in the case of the Bring-Jerrard form n = 5 leads. For equations of the fifth degree, only equations up to the fourth degree need to be solved. For equations of the fifth degree, the method, which Jerrard was not aware of, was found by Erland Samuel Bring in 1786. The Bring-Jerrard form for 5th degree equations was used by Charles Hermite for solving the 5th degree equation using elliptic module functions.

^ Rüdiger Thiele: Leonhard Euler, BSB BG Teubner Verlagsgesellschaft, Leipzig, 1982, p. 103

Web links

Mathworld - Quintic Equation - more details on methods for solving fifth degree equations.
Abels and Galois' Essays - the originals are available online.
Solving the Quintic with Mathematica - Fifth Degree Equation Poster.
Solving Solvable Quintics - a method for solving solvable equations of the fifth degree according to David S. Dummit. (PDF file; 93 kB)
Polynomial Transformations of Tschirnhaus, Bring and Jerrard - an essay by Victor S. Adamchik & David J. Jeffrey on polynomial transformations. (PDF file; 81 kB)

[1] Young, GP: Solution of Solvable Irreducible Quintic Equations, Without the Aid of a Resolvent Sextic. In: Amer. J. Math. Volume 7, pages 170-177, 1885.

[2] Runge, C .: About the solvable equations of the form . ${\ displaystyle x ^ {5} + ux + v = 0}$ In: Acta Math. Volume 7, pages 173-186, 1885, doi : 10.1007 / BF02402200 .