Suppose $K/\mathbb Q$ is a Galois extension of degree $3$. I know that this implies: $K/\mathbb Q$ is a splitting field of a separable polynomial $f(x)$ with coefficients in $\mathbb Q$. I can see why $\deg f(x) \neq 1$ and $\neq 2$. I think it can be that $\deg f(x) \ge 4$, but I also think that this can only happen if $f(x)$ already has roots in $\mathbb Q$; I am not sure though.
Can we extract from $f(x)$ a degree $3$ polynomial whose splitting field is $K$? Anyhow, does the question in the title have a positive answer?