Infinite Free Products of Profinite Groups as Absolute Galois Groups

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It is known that the free product of finitely many absolute Galois groups of fields is the absolute Galois group of a field. We give examples of free products of infinitely many nontrivial absolute Galois groups of fields which are not isomorphic to the absolute Galois group of any field. On the other hand, we give examples of free products of infinitely many nontrivial absolute Galois groups of fields which are the absolute Galois group of a field.