Quintic function

An equation of the fifth degree or quintic equation in mathematics, a polynomial of degree five. It is of the form

Where the coefficients and elements of the body (typically the rational, real or complex numbers ), are with. We then speak of an equation ( or a polynomial ) "over" this body.

• 3.1 Notes and references

History

The dissolution of polynomial equations by finite root expressions ( radicals) is an old problem. After Gerolamo Cardano in 1539 had published solutions to the general equations up to degree 4, efforts have focused on the solution of the general equation of the fifth degree. 1771 was Gianfrancesco Malfatti as a first approach, which only in the case of solubility, however, works by root expressions. Paolo Ruffini published in 1799 a fragmented evidence of the indissolubility of the general equation of degree 5. Since Ruffini used unfamiliar arguments for those times that are assigned to the group theory today, his evidence was not initially accepted. 1824 Niels Henrik Abel managed a complete proof that the general quintic equation is not solvable by radicals (Theorem of Abel- Ruffini ). In the Galois theory, the proof can be shortened represent as follows: The Galois group of the general equation of degree n has the Alternating group as a factor, and this group is just for (see icosahedral ), not resolvable. Charles Hermite succeeded in 1858, the general equation of the fifth degree in Jacobi theta functions ( but of course not in radicals) resolve.

Separable equations of the fifth degree

Some quintic equations can be solved with roots, for example, which can be factorized in the form. Other equations such as can not be solved by roots. Évariste Galois developed around 1830 methods to determine whether a given equation in roots is solvable (see Galois theory ). Based on these fundamental results proved George Paxton Young, and Carl Runge in 1885 an explicit criterion for deciding whether a given quintic equation with roots solvable (cf. the work of Lazard for a modern approach ). They showed that an irreducible quintic with rational coefficients in Bring- Jerrard form

Exactly then is solvable with roots, if they form the

With rational and has. In 1994, Blair Spearman and Kenneth S. Williams found the presentation

For. The relationship between the two parameterizations can be obtained by the equation

With

Be made ​​. In the case of the negative square root is obtained with the first parameterization, the second with a positive square root, with a suitable scaling. Therefore, it is a necessary (but not sufficient) condition for a separable quintic equation of the form

With rational, and that the equation

Has a rational solution.

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

Examples of separable equations of the fifth degree

An equation is solvable in radicals if its Galois group is a solvable group. For equations of degree its Galois group is a subgroup of the symmetric group of permutations of elements.

A simple example of a releasable equation is the Galois F ( 5) of the permutations "(1 2 3 4 5 )" and " ( 1 2 4 3 ) ' is generated; the only real root is

However, the solutions can also be much more complex. For example, the equation has the Galois group D (5 ) of which "(1 2 3 4 5 )" and " (1 4) (2 3)" is generated, and the solution requires written out about 600 symbols.