On the descending central sequence of absolute Galois groups

Let $p$ be an odd prime number and $F$ a field containing a primitive $p$th root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group $G_F$ of $F$. Namely, the third subgroup $G_F^{(3)}$ in the descending $p$-central sequence of $G_F$ is the intersection of all open normal subgroups $N$ such that $G_F/N$ is 1, $\mathbb{Z}/p^2$, or the modular group $M_{p^3}$ of order $p^3$.

💡 Research Summary

The paper investigates the structure of the descending p‑central series of absolute Galois groups. For an odd prime p and a field F containing a primitive p‑th root of unity, the authors prove that the third term G_F^{(3)} of the descending p‑central series is precisely the intersection of all open normal subgroups N of G_F for which the quotient G_F/N is isomorphic to one of three groups: the trivial group, the cyclic group ℤ/p², or the non‑abelian group M_{p³} of order p³ and exponent p². This result provides a new, sharp restriction on possible finite quotients of absolute Galois groups beyond the well‑known case p=2.

The proof rests on a cohomological framework. The authors introduce the notion of a profinite group of “Galois relation type” relative to a power q=p^d. Such a group satisfies three conditions: (i) the kernel of the cup‑product map H¹(G)⊗H¹(G)→H²(G) is generated by pure tensors ψ⊗ψ′; (ii) there exists a distinguished element ξ∈H¹(G) with ψ∪ξ+β_G(ψ)=0 for every ψ, where β_G is the Bockstein homomorphism; (iii) the natural restriction maps H¹(G,ℤ/q)→H¹(G,ℤ/p^i) are surjective for all i≤d. Using Kummer theory and the Merkurjev–Suslin theorem, they verify that any absolute Galois group G_F of a field containing a primitive q‑th root of unity satisfies these axioms.

A key technical tool is the construction of an auxiliary abelian group Ω(G) and a homomorphism Λ_G:Ω(G)→H²(G,ℤ/q) extending the cup‑product while incorporating the Bockstein map. The kernel of Λ_G is shown to be generated by “simple‑type” elements, which correspond to certain finite index normal subgroups N of G (called distinguished subgroups) whose quotients have exponent dividing q³. The authors prove that for groups of Galois relation type, G^{(3)} equals the intersection of all distinguished subgroups. Translating this into concrete group‑theoretic terms yields the main theorem: for p odd, G_F^{(3)} is the intersection of all open normal N with G_F/N≅1, ℤ/p², or M_{p³}.

The paper also revisits the case p=2. By adapting the same machinery, the authors recover the classical result that G_F^{(3)} is the intersection of kernels of quotients isomorphic to 1, ℤ/2, ℤ/4, or the dihedral group D₄, and they show that the factor ℤ/2 can be omitted unless F is Euclidean. Further sections explore the structure of the quotient G_F/G_F^{(3)}, provide “automatic realization” corollaries, and present explicit examples demonstrating that every group appearing in the list indeed occurs as a quotient. Overall, the work offers a cohomology‑driven, unified approach to understanding higher layers of absolute Galois groups, extending known results and opening avenues for further classification problems.