The Sylow subgroups of the absolute Galois group \(\mathrm{Gal}(\mathbb{Q})\). (English) Zbl 1334.12003
Summary: We describe the \(\ell\)-Sylow subgroups of \(\mathrm{Gal}(\mathbb{Q})\) for an odd prime \(\ell\), by observing and studying their decomposition as \(F \rtimes \mathbb{Z}_\ell\), where \(F\) is a free pro-\(\ell\) group, and \(\mathbb{Z}_\ell\) are the \(\ell\)-adic integers. We determine the finite \(\mathbb{Z}_\ell\)-quotients of \(F\) and more generally show that every split embedding problem of \(\mathbb{Z}_\ell\)-groups for \(F\) is solvable. Moreover, we analyze the \(\mathbb{Z}_\ell\)-action on generators of \(F\).
MSC:
| 12F10 | Separable extensions, Galois theory |
| 11R32 | Galois theory |
| 12F12 | Inverse Galois theory |
| 20E18 | Limits, profinite groups |
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