TITLE: Bound for the degree
QUESTION [5 upvotes]: Let $K$ be a perfect field and let $f\in K[x]$ be a monic irreducible polynomial of degree $n$.
Denote by $\alpha,\beta$ two distinct roots of $f$. 
Is the following bound true?
$$
[K(\alpha-\beta):K]\geq \frac n2
$$
If not, does someone know a similar bound (if it exists)?

REPLY [5 votes]: This bound does not hold. Consider the Artin-Schreier polynomial, $$f(X)=X^p-X+1\in\mathbb{F}_p[X]$$
Note that $f(\alpha)=0\implies f(\alpha+1)=0$. Take $\beta=\alpha+1$ to obtain a contradiction to the proposed bound.