Let me ask something about separable polynomials. I found the following definitions:
A polynomial over $F$ is called separable if it has no multiple roots (i.e. all its roots are distinct). (Dummit & Foote)
A polynomial over $F$ is called separable if every irreducible factor has not multiple roots (the irreducible factors may not be distinct. (Rotman)
My question is: Are these two definitions equivalent? Let me give an example which looks strange:
Lets take the polynomial $f(x)= (x^2-2)^n \in \Bbb{Q}[x]$, with $n\in \Bbb{N}^{\geq 2}$. Then, according to def. 1 $$f(x)=(x-\sqrt2)^n(x+\sqrt2)^n $$ so the roots are multiple, so it is not separable over $\Bbb{Q}$. But, according to def. 2 $$f(x)= \underbrace{(x-\sqrt2)(x+\sqrt2)\cdots (x-\sqrt2)(x+\sqrt2)\,}_\text{$n$ times} $$ so no one of the irreducible factors has root with multiplicity $=1$. So, it is separable over $\Bbb{Q}$.
How can this happen? Do I misunderstand something?