Two different definitions of separable polynomial

This is from A field guide to Algebra by Antoine Chambert Loir.

A polynomial $P \in K[X]$ is separable if and only if its roots are in an algebraic closure of $K$ are simple.

Here is another definition.

An irreducible polynomial $f \in F [X]$ is separable if $f$ has no repeated roots in a splitting field. It is called inseparable otherwise. Note that if $f$ is not necessarily irreducible, then we call $f$ separable if each of its irreducible factors is separable.

Why two different definitions?

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Are they realy different? Consider an algebraic closure of the splitting field, resp. the splitting field within a given algebraic closure. –  Hagen von Eitzen Feb 7 '13 at 12:26
$f (X) = (X − 1)^2 (X − 2) \in \mathbb Q$. This polynomial is not seperable according to def 1 because 1 is not a simple root of $f(X)$ . –  Mohan Feb 7 '13 at 12:30
@HagenvonEitzen, isn't $(x-1)^2 \in \mathbf{Q}[x]$ separable with respect to the second definition, but not with respect to the first one? –  Andreas Caranti Feb 7 '13 at 12:31
Second def. is about irreducible polynomials. –  Berci Feb 7 '13 at 12:44
When a polynomial is not irreducible two definitions are different. –  Mohan Feb 7 '13 at 12:54

2 Answers

See http://en.wikipedia.org/wiki/Separable_polynomial for a good discussion of this.

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Thanks for the link. –  Mohan Feb 7 '13 at 12:31

By second definition we can say every polynomial is separable and thus there is no question about inseparability. So there no further work to do. By the first definition, we an study both separability and inseparability as well as multiplicity and splitting field. So most of cases authors' consider the separability of irreducible polynomial as it support both the definitions....

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