Radical extensions

Question

Suppose we have $$ K \subset K(a_1) \subset K(a_1, a_2) \subset K(a_1, a_2, a_3) $$ such that $a_j^{p_j}\in K_{j-1}$: a radical extension in other words.

I am having trouble understanding why for example the extension $K(a_1, a_2):K(a_1)$ need not be radical. The criteria "$a_j^{p_j}\in K_{j-1}$" still holds over this extension right? So then, how come intermediate extensions need not be radical?

Thank you.

Answer 1

Why do you think that the chain of subfields that you've described isn't radical?

Definition. A finite extension $L/K$ is radical if you can find a finite chain of subfields $K=L_0\subset L_1\subset\cdots\subset L_n=L$ such that $L_i=L_{i-1}(a_i)$, and $a_i^{p_i}\in K_{i-1}$, for some $p_i>1$.

Let $K_1=K(a_1)$. Then you're saying that the extension $K_1(a_2)/K_1$, where $a_2^{p_2}\in K_1$, for some $p_2>1$, need not be radical? It certainly seems to satisfy the definition.