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separable polynomial
In mathematics, a polynomial ''P''(''X'') over a given field ''K'' is separable if its roots are distinct in an algebraic closure of ''K'', that is, the number of its distinct roots is equal to its degree.〔S. Lang, Algebra, p. 178〕
This concept is closely related to square-free polynomial. If ''K'' is a perfect field then the two concepts coincide. In general, ''P''(''X'') is separable if and only if it is square-free over any field that contains ''K'',
which holds if and only if ''P''(''X'') is coprime to its formal derivative ''P''′(''X'').
==Older definition==
In an older definition, ''P''(''X'') was considered separable if each of its irreducible factors in K() is separable in the modern definition〔N. Jacobson, Basic Algebra I, p. 233〕 In this definition, separability depended on the field ''K'', for example, any polynomial over a perfect field would have been considered separable. This definition, although it can be convenient for Galois theory, is no longer in use.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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