Editing Separable extension (section)

==Informal discussion==
An arbitrary polynomial {{math|''f''}} with coefficients in some field {{math|''F''}} is said to have ''distinct roots'' or to be [[square-free polynomial|square-free]] if it has {{math|deg ''f''}} roots in some [[extension field]] <math>E\supseteq F</math>. For instance, the polynomial {{math|1=''g''(''X'') = ''X''<sup>&thinsp;2</sup> − 1}} has precisely {{math|1=deg&thinsp;''g'' = 2}} roots in the [[complex plane]]; namely {{math|1}} and {{math|−1}}, and hence ''does have'' distinct roots. On the other hand, the polynomial {{math|1=''h''(''X'') = (''X'' − 2)<sup>2</sup>}}, which is the square of a non-constant polynomial ''does not'' have distinct roots, as its degree is two, and {{math|2}} is its only root.

Every polynomial may be factored in linear factors over an [[algebraic closure]] of the field of its coefficients. Therefore, the polynomial does not have distinct roots if and only if it is divisible by the square of a polynomial of positive degree. This is the case if and only if the [[polynomial greatest common divisor|greatest common divisor]] of the polynomial and its [[formal derivative|derivative]] is not a constant. Thus for testing if a polynomial is square-free, it is not necessary to consider explicitly any field extension nor to compute the roots.

In this context, the case of irreducible polynomials requires some care. A priori, it may seem that being divisible by a square is impossible for an [[irreducible polynomial]], which has no non-constant divisor except itself. However, irreducibility depends on the ambient field, and a polynomial may be irreducible over {{math|''F''}} and reducible over some extension of {{math|''F''}}. Similarly, divisibility by a square depends on the ambient field. If an irreducible polynomial {{math|''f''}} over {{math|''F''}} is divisible by a square over some field extension, then (by the discussion above) the greatest common divisor of {{math|''f''}} and its derivative {{math|''f''{{′}}}} is not constant. Note that the coefficients of {{math|''f''{{′}}}} belong to the same field as those of {{math|''f''}}, and the greatest common divisor of two polynomials is independent of the ambient field, so the greatest common divisor of {{math|''f''}} and {{math|''f''{{′}}}} has coefficients in {{math|''F''}}. Since {{math|''f''}} is irreducible in {{math|''F''}}, this greatest common divisor is necessarily {{math|''f''}} itself. Because the degree of {{math|''f''{{′}}}} is strictly less than the degree of {{math|''f''}}, it follows that the derivative of {{math|''f''}} is zero, which implies that the [[characteristic of a field|characteristic]] of the field is a prime number {{math|''p''}}, and {{math|''f''}} may be written
:<math>f(x)= \sum_{i=0}^ka_ix^{pi}.</math>

A polynomial such as this one, whose formal derivative is zero, is said to be ''inseparable''. Polynomials that are not inseparable are said to be ''separable''. A ''separable extension'' is an extension that may be generated by ''separable elements'', that is elements whose minimal polynomials are separable.