Bring radical

In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial x 5 + x + a . {\displaystyle x^{5}+x+a.} The Bring radical defines x {\displaystyle x} as an algebraic function of a {\displaystyle a} . It is the simplest algebraic function that cannot be expressed in terms of radicals. 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 BR ⁡ ( a ) . {\displaystyle \operatorname {BR} (a).} For real argument, it is odd, monotonically decreasing, and unbounded, with asymptotic behavior BR ⁡ ( a ) ∼ − a 1 / 5 {\displaystyle \operatorname {BR} (a)\sim -a^{1/5}} for large a {\displaystyle a} .