rezolvabil

Rezolvabil, or solvable in English, is a mathematical property describing the possibility of solving a problem or structure by a finite sequence of explicit, elementary steps within a given framework.

In group theory, a group G is solvable if it admits a finite chain of subgroups G = G0 ⊳ G1 ⊳ ... ⊳ Gr = {e} where each Gi+1 is normal in Gi and the quotient Gi/Gi+1 is abelian. Equivalently, the derived series G^(0) = G, G^(n+1) = [G^(n), G^(n)] reaches the trivial group after finitely many steps. Examples include finite abelian groups and p-groups. Among symmetric groups, S_n is solvable for n ≤ 4, but not for n ≥ 5.

In the context of polynomial equations, solvability by radicals means that the roots can be expressed using a finite sequence of additions, multiplications, and extraction of roots. Galois theory links solvability by radicals to the solvability of the Galois group of the polynomial: a polynomial is solvable by radicals over a field if and only if its Galois group is solvable. As a consequence, cubic and quartic equations are solvable by radicals, while the general quintic is not solvable in general.

The concept extends to other areas of algebra and analysis where a finite procedure or transformation yields solutions. Rezolvabil thus denotes that the problem, equation, or algebraic object can be handled within a finite, well-defined framework.