Definition of solvable

The word "solvable" in mathematics originated in the late 19th century, particularly in the context of solving equations. The prefix "sol" in the word comes from the Latin word "solvere," meaning "to loose," "to release," or "to dissolve." By the early 20th century, mathematicians began applying the term "solvable" to sets of instructions or equations that could be solved using a set of predefined steps or algorithms. This concept is closely related to the notion of a group, which is a set of elements with certain properties that can be represented geometrically or algebraically. Solvable groups are a subclass of finite groups for which it is possible to construct a finite sequence of substeps that will reduce them to a simpler form. In practical terms, this means that there exist efficient procedures or algorithms for solving problems related to solvable groups or equations, making them more manageable and less complex than unsolvable ones. This distinction has important applications in various mathematical fields, such as algebra, number theory, and cryptography, where it is crucial to be able to efficiently solve certain problems or equations.