I am currently study Field Arithmetic written by M.Fried and M.Jarden.
Here is one of the theorem as following which is Propostion 13.4.6 in that book:
Every field K has a regular extension F which is PAC and Hilbertian
The first part of the proofs goes as follows:
Wellorder the (absolutely irreducible) varieties of dimension at least 1 that are defined over K in a transfinite sequence $\{ V_\alpha | \alpha < m \}$ for some ordinal m. Use transfinite induction to define, for each $\beta < m$, a function field $F_\beta$ for $V_\beta$ which is algebraically independent from $\prod_{\alpha <\beta}F_\alpha$ (the composite of the $F_\alpha$'s with $\alpha < \beta$) over K. Since $F_\beta$ is a regular extension of K, the field $\prod_{\alpha \leq \beta}F_\alpha$ is a proper finitely generated regular extension of $\prod_{\alpha <\beta}F_\alpha$ ( Corollary 2.6.8(a)). It follows that $K_1=\prod_{\alpha <m}F_\alpha$ is Hilbertian (Lemma 13.4.5) and regular over K (Lemma 2.6.5(d)).
My question is that why every absolutely irreducible variety defined over $K$ has a $K_1$-rational point?
Any hints or answers are welcomed !