Condition for having order 2 elements in Galois group for polynomials over $\mathbb{Q}$ Suppose we have $f \in \mathbb{Q}[X]$, with only real roots. Then the complex conjugation is not an automorphism, but is this enough to say that there exist no order two elements in $\text{Gal}(f)$?
The case I was studying is $f = x^3-4x+2$, I found that it has $3$ real roots. But can I now say that there is no order $2$ element in the Galois group?

By Lulu:
$f(x) = x^2 - 2$ is a counter example, as we have the automorphism: $\sigma(a+b\sqrt{2}) = a - b\sqrt{2}$.
