nLab solvable polynomial

A polynomial ϕ\phi over an integral domain KK (or equivalently a polynomial equation, which is the generality that one should consider if KK is only an integral cancellable rig) is solvable if all of its roots in an algebraic closure (or any other sufficiently large splitting field) of KK are expressible using elements of KK, the field operations, and extraction of roots.

The basic result of elementary Galois theory is that ϕ\phi is solvable over KK if and only if its Galois group over KK is a solvable group (whence the name of the latter).

Last revised on September 16, 2016 at 00:01:00. See the history of this page for a list of all contributions to it.