RootOfP

RootOfP is a notation used to denote a root of a polynomial P, primarily in algebra, symbolic computation, and computer algebra systems. The term RootOf(P) (sometimes written as RootOfP) refers to a formal object representing a root of P without requiring an explicit closed form. For a field F and a polynomial P in F[x] with degree at least one, a root α in an algebraic closure of F satisfies P(α) = 0. The RootOf(P) object stands for such a root, and when several roots exist, a second index i may be used, as in RootOf(P, i), to designate the i-th root in a fixed, canonical ordering.

RootOf objects enable algebraic manipulations without solving the polynomial by radicals. They participate in symbolic operations such as factorization, elimination, and the computation of resultants, and they can be numerically approximated to arbitrary precision by evaluating RootOf(P, i) with numerical methods. The exact indexing of i is defined by the system or algorithm being used, often based on a prescribed ordering of roots by real parts or other canonical criteria.

In theory, RootOf(P) is closely related to the splitting field of P over F. If P is irreducible over F, any RootOf(P, i) generates an algebraic extension of F of degree equal to deg P. The concept is widely used in the study of algebraic numbers, Galois theory, and computational algebra.

Example: for P(x) = x^2 − 2 over Q, the two RootOf objects correspond to the two real roots, ±sqrt(2); in practice, they may be represented numerically without explicit radical expressions.