solvable polynomial

A polynomial $\phi$ over an integral domain $K$ (or equivalently a polynomial equation, which is the generality that one should consider if $K$ 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 $K$ are expressible using elements of $K$, the field operations, and extraction of roots.

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

Last revised on September 15, 2016 at 20:01:00. See the history of this page for a list of all contributions to it.