In my textbook, I see the mentioning of the cubic root of an element in a field with the notation $\sqrt[3]{a}$ for $a \in F$ with $F$ being a field.
I am not sure if I intepreted this correctly: $\sqrt[3]{a}$ are referring to the three roots of the polynomial $x^3 - a$ in its splitting field over $F$.
Please confirm if my understanding was correct. Thank you.
It usually refers to one of the roots, chosen either arbitrarily or following some general principle. For instance, for real numbers, we choose $\sqrt[3]{a}$ to denote the real root of $x^3-a$, not any of the two other complex roots. On the other hand, for a field like $\Bbb F_7$, there is no significant difference between the three choices for $\sqrt[3]{2}$, so we just pick one of them.