# 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