Just want to make it clear for the Defintion of Splitting Field

Question

If $E = \mathbb{Q}(\alpha_1, \alpha_2, \alpha_3)$ the splitting field of $f(x)$ over $\mathbb{Q}$.
Then, does that have to be the case which $\alpha_1$, $\alpha_2$ and $\alpha_3$ are distinct roots of $f(x)$ ?
And also, does that have to be the case which $\alpha_1$, $\alpha_2$ and $\alpha_3$ are the only roots of $f(x)$ ?

Answer 1

No, it only means that $f(x)$ splits into linear factor in $E$ and in no proper subfield of $E.$

For example, $\mathbb Q(\sqrt 2,\sqrt 3)$ is the splitting field of $f(x)=x^4-10x^2+1.$ Neither $\sqrt 2,\sqrt 3$ are roots of $f(x).$ (The roots are $\pm\sqrt 2\pm \sqrt 3.$)