Regular split embedding problems over complete valued fields

We give an elementary self-contained proof of the following result, which Pop proved with methods of rigid geometry. Theorem: Let L0/K0 be a finite Galois extension of complete discrete valued fields. Let t be a transcendental element over K0, let K = K0(t) and L = L0(t). Then each finite split embedding problem G → G(L/K) over K has a solution field F which is regular over L0. This gives a new proof of the theorem of Fried-Pop-Völklein: Theorem: The absolute Galois group of a countable separably Hilbertian PAC field K is the free profinite group on countably many generators