Solvability

Solvability is the property of a problem, equation, or system that allows for its solution to be found, typically in a finite sequence of well-defined steps or through a constructive method. In mathematics, the term is often used to indicate whether a problem can be solved in principle and how such a solution might be obtained.

In algebra, solvability is commonly discussed for polynomial equations. An equation is solvable by radicals if its solutions can be expressed using a finite combination of additions, multiplications, and taking roots of the coefficients. Galois theory connects solvability by radicals to the structure of the equation’s symmetry group; the general polynomial of degree five or higher is not solvable by radicals, although many specific cases are solvable.

In group theory, a group is called solvable if it has a chain of subgroups in which successive quotient groups are abelian. This concept generalizes the idea of reducing a problem to simpler components. The solvability of a group is closely linked to the solvability of polynomials via Galois theory: equations whose Galois group is solvable are solvable by radicals.

In logic and computer science, solvability is often framed in terms of decidability and computability. A problem is decidable if an algorithm exists that terminates with a correct yes-or-no answer for every instance. Many problems are undecidable, illustrating limits on what can be algorithmically solved.