A polynomial is solvable if and only if its corresponding Galois group is "solvable". Solvable groups are motivating primarily by the problem of determining if

A polynomial is solvable if and only if its corresponding Galois group is "solvable". Solvable groups are motivating primarily by the problem of determining if polynomials are solvable. In general, it takes some work to know if a group is solvable, and I don't think there is a way to determine that immidiatly. The most charitable reading of the statement is that there are relativly few groups of small order. This means that for polynomials of relativly small degree you have probably seen their Galois group before so can go off of memory.