separable
An irreducible polynomial with coefficients
in a field is separable if factors into distinct linear factors over a splitting field
of .
A polynomial with coefficients in is separable if each irreducible
factor of in is a separable polynomial.
An algebraic field extension is separable if, for each , the minimal polynomial of over is separable. When has characteristic zero, every algebraic extension of is separable; examples of inseparable extensions include the quotient field over the field of rational functions in one variable, where has characteristic
.
More generally, an arbitrary field extension is defined to be separable if every finitely generated intermediate field extension has a transcendence basis such that is a separable algebraic extension of .