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Concept
Separable polynomial
Résumé
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 distinct roots is equal to the degree of the polynomial.
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 D P(X).
In an older definition, P(X) was considered separable if each of its irreducible factors in K[X] is separable in the modern definition. 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.
Separable polynomials are used to define separable extensions: A field extension K ⊂ L is a separable extension if and only if for every α in L which is algebraic over K, the minimal polynomial of α over K is a separable polynomial.
Inseparable extensions (that is, extensions which are not separable) may occur only in positive characteristic.
The criterion above leads to the quick conclusion that if P is irreducible and not separable, then D P(X) = 0.
Thus we must have
P(X) = Q(X p)
for some polynomial Q over K, where the prime number p is the characteristic.
With this clue we can construct an example:
P(X) = X p − T
with K the field of rational functions in the indeterminate T over the finite field with p elements. Here one can prove directly that P(X) is irreducible and not separable. This is actually a typical example of why inseparability matters; in geometric terms P represents the mapping on the projective line over the finite field, taking co-ordinates to their pth power. Such mappings are fundamental to the algebraic geometry of finite fields. Put another way, there are coverings in that setting that cannot be 'seen' by Galois theory.
Source officielle
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