Here is the definition I have:
For a field $F,$ let $\bar{F}$ denote its algebraic closure. An element $\alpha \in \bar{F}$ is radical over $F$ if there exist an integer $n \geq 1$ and elements $t_{1}, ... , t_{2} \in \bar{F}$ such that:
$(1) \alpha \in F(t_{1}, ... , t_{n})$
$(2)$ For each i, $1 \leq i \leq n-1,$ there exists an integer $d_{i} \geq 1$ with $t_{i+1}^{d_{i}} \in F(t_{1}, ... , t_{i})$
I am new to the subject and I do not understand the definition and we are using a book called "Algebra" by Michael Artin which even does not contain the definition. Could anyone give me an example for applying the definition specifically $(2)$ in it?
This definition means basically that $\alpha$ can be expressed by applying the four field operations $+,-,\cdot,/$ and the operation of 'taking $k$th root', starting out from elements of $F$.
For example, $\frac{\sqrt[3]{\sqrt 5-\sqrt[4]7}+\sqrt2}{\sqrt[19]{6-\sqrt3}}$ is a radical element over $\Bbb Q$.
Note that, however, the $k$th root operation is not well defined as an operation, because $x^k=a$ typically has more than $1$ (specifically, $k$) solutions in an algebraically closed field.
Also note the famous result of Galois, which says that there is no general formula with root signs and field operations on the coefficients to solve polynomial equations of order $\ge5$.
Moreover, it can be shown using specific quintic polynomials, that not every algebraic numbers are radical.