Risolvibile

Risolvibile is an Italian adjective that means capable of being solved, resolved, or settled. In mathematical and logical language it is used to describe problems, equations, or structures for which a solution can, in principle, be found or determined, often by a specified method.

In mathematics, risolvibile is commonly used in several contexts. An equation or a system of equations is risolvibile if a method exists to obtain its solution. For polynomials, a classical phrase is “risolvibile per radicali,” referring to solvability by radicals through a finite sequence of algebraic operations. Linear equations are typically risolvibili when the system has a solution, and, in the case of a nonsingular coefficient matrix, the solution is unique. Differential equations can be described as risolvibile when their general or particular solutions can be expressed explicitly or derived algorithmically.

In abstract algebra, the term is prominent in the notion of solvable groups. A group is risolvibile (solvable) if it possesses a finite derived series whose successive quotients are abelian, equivalently, if the group can be broken down into abelian pieces through a series of standard constructions. This concept is central to Galois theory, where the solvability of polynomials by radicals is linked to the structure of associated groups.

In logic and computer science, risolvibile is sometimes used to indicate that a problem admits an algorithmic solution. More precise terminology distinguishes decidibile problems (yes/no answers for all inputs) from problems that may be partially solvable or semi-decidable.

Etymology traces to the verb risolvere (to solve) with the suffix -ibile, yielding the sense of “able to be solved.”