# nLab 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).

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