If $K$ is a field of charateristic $0$ then every irriducible polynomial over $K$ is separable over $K$

Question

Let the seguent propositions:

Lemma $1$:

A polynomial $f \not=0$ over a field $K$ has a multiple zero in a splitting field if and only if $f$ and $Df$ have a common factor of degree $\ge1$

Lemma $2$:

If $K$ is a field of charateristic $0$ then every irriducible polynomial over $K$ is separable over $K$

Proof Lemma $2$: An irredubicible polynomial $f$ over $K$ is inseparable if and only if Lemma $1$. If so, then since $f$ is irreducible and $Df$ has smaller degree than $f$. We must have $Df=0$

I don't understand last implication. If I take in $\mathbb{Q}$ the polynomial $x^2+1$ is irreducible. I know that $Df=2x$ and $HCF(f,Df)=1$ and $deg (HCF(f,DF))=deg(1)=0$. In this example I don't have that $Df=0$

Where is the mistake that i'm doing?

Answer 1

1) $\;f\;$ is irreducible, so it cannot have non-trivial factors.

2) If $\;f\,,\,\,f'\;$ have a common factor, then $\;f\;$ is reducible unless $\;f'=0\;$

Both points above mean, at least, that if $\;f'\neq 0\implies f\;$ is separable