In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial
The Bring radical of a complex number a is either any of the five roots of the above polynomial (it is thus multi-valued), or a specific root, which is usually chosen such that the Bring radical is real-valued for real a and is an analytic function in a neighborhood of the real line. Because of the existence of four branch points, the Bring radical cannot be defined as a function that is continuous over the whole complex plane, and its domain of continuity must exclude four branch cuts.
George Jerrard showed that some quintic equations can be solved in closed form using radicals and Bring radicals, which had been introduced by Erland Bring.
In this article, the Bring radical of a is denoted For real argument, it is odd, monotonically decreasing, and unbounded, with asymptotic behavior for large .
Normal forms
The quintic equation is rather difficult to obtain solutions for directly, with five independent coefficients in its most general form:
The various methods for solving the quintic that have been developed generally attempt to simplify the quintic using Tschirnhaus transformations to reduce the number of independent coefficients.
Principal quintic form
The general quintic may be reduced into what is known as the principal quintic form, with the quartic and cubic terms removed:
If the roots of a general quintic and a principal quintic are related by a quadratic Tschirnhaus transformation
the coefficients α and β may be determined by using the resultant, or by means of the power sums of the roots and Newton's identities. This leads to a system of equations in α and β consisting of a quadratic and a linear equation, and either of the two sets of solutions may be used to obtain the corresponding three coefficients of the principal quintic form.
This form is used by Felix Klein's solution to the quintic.
Bring–Jerrard normal form
It is possible to simplify the quintic still further and eliminate the quadratic term, producing the Bring–Jerrard normal form:
Using the power-sum formulae again with a cubic transformation as Tschirnhaus tried does not work, since the resulting system of equations results in a sixth-degree equation. But in 1796 Bring found a way around this by using a quartic Tschirnhaus transformation to relate the roots of a principal quintic to those of a Bring–Jerrard quintic:
The extra parameter this fourth-order transformation provides allowed Bring to decrease the degrees of the other parameters. This leads to a system of five equations in six unknowns, which then requires the solution of a cubic and a quadratic equation. This method was also discovered by Jerrard in 1852, but it is likely that he was unaware of Bring's previous work in this area. The full transformation may readily be accomplished using a computer algebra package such as Mathematica or Maple. As might be expected from the complexity of these transformations, the resulting expressions can be enormous, particularly when compared to the solutions in radicals for lower degree equations, taking many megabytes of storage for a general quintic with symbolic coefficients.
Regarded as an algebraic function, the solutions to
involve two variables, d1 and d0; however, the reduction is actually to an algebraic function of one variable, very much analogous to a solution in radicals, since we may further reduce the Bring–Jerrard form. If we for instance set
then we reduce the equation to the form
which involves z as an algebraic function of a single variable , where . This form is required by the Hermite–Kronecker–Brioschi method, Glasser's method, and the Cockle–Harley method of differential resolvents described below.
An alternative form is obtained by setting
so that
where . This form is used to define the Bring radical below.
Brioschi normal form
There is another one-parameter normal form for the quintic equation, known as Brioschi normal form
which can be derived by using the rational Tschirnhaus transformation
to relate the roots of a general quintic to a Brioschi quintic. The values of the parameters and may be derived by using polyhedral functions on the Riemann sphere, and are related to the partition of an object of icosahedral symmetry into five objects of tetrahedral symmetry.
This Tschirnhaus transformation is rather simpler than the difficult one used to transform a principal quintic into Bring–Jerrard form. This normal form is used by the Doyle–McMullen iteration method and the Kiepert method.