We give an elementary self-contained proof of the following result, which Pop proved with methods of rigid geometry.Theorem: Let L 0 /K 0 be a finite Galois extension of complete discrete valued fields.Let t be a transcendental element overThis 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.

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n i=1 G i is an absolute Galois group of a field of characteristic p.Mel'nikov [Mel, Thm. 1.4], proves the Theorem when rank(G i ) ≤ ℵ 0 , i = 1, . . . , n.His proof uses a theorem of Geyer [Gey]: Suppose M and L are Henselian fields with respect to rank 1 valuations and both are separable algebraic extensions of a countableHere G(K) is the absolute Galois group of K and "almost all" is used in the sense of the Haar measure of G(K). Mel'nikov's proof does not extend to the case of uncountable rank.Ershov [Ers, Thm. 3] proves part (a) of the theorem by a different method. Our proof simplifies that of Ershov. We take this opportunity to supply proofs to well known results which are not well documented in the literature.The Theorem is also a consequence of [Pop, Thm. 3.4]. As in the proof of Theorem 3.4, one may use Proposition 2.5 to obtain, in the terminology of [Pop, Ch. 3, §2], a Galois approximation of * n i=1 G i . Then one map apply [Pop, Thm. 3.4].