A central issue in modern Galois theory is the profinite inverse Galois problem, which asks how to characterize absolute Galois groups of fields among all profinite groups. While an answer to this question is unknown, even conjecturally, several necessary conditions for a profinite group to qualify as an absolute Galois group have been established. The most classical result in this direction is due to Artin and Schreier, who proved that every non-trivial finite subgroup of an absolute Galois group is cyclic of order 2. A much deeper necessary condition is the Bloch-Kato conjecture, now a theorem due to Voevodsky and Rost, which in particular implies that the mod p cohomology ring of an absolute Galois group of a field containing a primitive p-th root of unity is generated in degree 1 with relations in degree 2. In the lecture, we will discuss restrictions to the profinite inverse Galois problem coming from the embedding problem with abelian kernel. This is a joint work with Federico Scavia.

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Invited by Prof. Nikita Geldhauser