Checking regularity of field extension for Chatzidakis notes

I have been referencing Chatzidakis' notes on psuedofinite fields section 6.7, found here, and her explanation of the simple case of Duret's result that the theory of any pseudo-algebraically closed fields which are not separably closed has the independence property.

Let $F$ be a pseudofinite field. Chatzidakis states without proof that, given $a_i, c \in F$, $a_i$ distinct, and $c$ nonsquare, the extension $$L = F (t, \sqrt{t + a_1}, . . . , \sqrt{t + a_r} , \sqrt{c(t + a_{r+1})}, . . . , \sqrt{c(t + a_n)})$$ is regular; i.e. that $L \cap F^{alg} = F$. This is clear for $n=1$, but I'm lost beyond that. I've done some digging and been unable to find a reference for this statement; she mentions a more complete set of notes from a course she gave in Madrid, but they seem to no longer be online.

I'm interested in formulating a more general version of this statement (i.e. more than just square roots), but I've been unable to find Duret's result in English, nor can I follow enough of the content of Jarden and Fried's "Field Arithmetic" to tell what may be helpful to me. If anyone can explain how this condition is verified or direct me to a specific location in some source that explains it, I would really appreciate it. More generally, I'm looking for some framework I can use to evaluate whether a finitely generated extension in a pseudofinite (or more relevantly, perfect) field has this property of regularity.

To summarize my questions:

1. How does one show that $L \cap F^{alg} = F$ for the above extension?
2. What tools are available to show that a general finitely generated extension of a pseudofinite (perfect) field has this property?
3. Is there a English source available to reference, from Chatzidakis, Duret, or anywhere else, which explains how to answer 1 and/or 2?

Apologies in advance for my shaky forum etiquette; I tried to follow the guidelines, but this is my first post, so I imagine it won't be perfect.