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In [[mathematics]], an [[irreducible polynomial]] ''P''(''X'') is '''separable''' if its roots in an [[algebraic closure]] are [[distinct]] - that is ''P''(''X'') has distinct linear factors in some large enough [[field extension]]. This is the 'usual' case, since it turns out to be true if ''P'' is defined over a [[field (mathematics)|field]] ''K'' that is either (a) of [[characteristic]] 0, or (b) a [[finite field]]. This criterion is of technical importance in [[Galois theory]].
If ''P'' is not assumed irreducible the concept is of lesser importance, since repeated roots may then just reflect that ''P'' is not [[square-free polynomial|square-free]] . We can test for common factors of ''P''(''X'') and the [[derivative]] ''P''′(''X'') over any field, using the calculus formula: any repeated root will divide the [[highest common factor]] of ''P'' and ''P''′. This leads to the quick conclusion that if ''P'' is irreducible and ''not'' separable, then ''P''′(X) = 0. This is only possible as a characteristic ''p'' phenomenon: we must have ''P''(''X'') = ''Q''(X<sup>''p''</sup>) where the prime number ''p'' is the characteristic.
With this clue we can construct an example:
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