Embedding Problems Over Large Fields

Henselian implies large

“…The above Problems have positive solutions over fields K which are large fields, see e.g. [Pop96,Main Theorems A and B]. The large fields were introduced in loc.…”

“…Examples of large fields are the PAC fields, the complete fields like k..x//, the real /p-adically closed fields, and more generally the Henselian valued fields, the p-fields, etc. See Pop [Pop96] for more about large fields.…”

On the Characterization of Hilbertian Fields

“…any finite embedding problem has a proper solution). These results are generalized for a field with an arbitrary characteristic by Pop [5].…”

Free Product Decomposition of Galois Groups of Number Fields

“…If T 0 = S ∞ is the set archimedean primes, this is a result of Fried-Haran-Völklein [1] and in general it is proven by Pop [5]. We also would like to mention a result of Neukirch, [3] Satz (12.2), which says that finitely many decomposition groups G P (k(p)|k) form a free pro-pproduct inside G(k(p)|k).…”