Bring–Jerrard form

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 partially undefined), or a specific root, which is usually chosen in order that the Bring radical is a function of a, which is real-valued when a is real, 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 ${\displaystyle \operatorname {BR} (a).}$

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.

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