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Concept
Bring radical
Summary
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 .
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.
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.
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