×

The ramification filtration of the Galois group of a local field. (Russian) Zbl 0853.11096

Let \(K\) be a local complete discretely valued field with a perfect residue field of characteristic \(p\). The Galois group \(G= \text{Gal} (K_{\text{sep}}/ K)\) has the ramification filtration. The essential part is that this filtration coincides with the group \(I= \text{Gal} (K_{\text{sep}}/ K_{\text{tr}})\) over the maximal tamely ramified extension over \(K\). The groups \(G\) and \(I\) can be described explicitly. The main problem discussed in the paper is how to describe the ramification filtration in purely group-theoretic terms. The author assumes that \(\text{char} (K)= p\) and \(k= F_p\). Then \(I\) is a profinite-\(p\)-free group. An explicit description is given of the ramification filtration in the quotient-group \(I/I^p C_p\) where \(C_p\) is the subgroup generated by the commutators of order \(>p-1\). To get this result the author has developed a non-abelian version of the Artin-Schreier theory which works for arbitrary extensions with the Galois group of period \(p\) and nilpotency class \(<p\). In this non-abelian Artin-Schreier theory the Galois groups can be described using the nilpotent Lie algebras (of the same nilpotency class). The author hopes that this approach may work also without the nilpotency condition for fields \(K\) of zero characteristic.

MSC:

11S20 Galois theory
PDF BibTeX XML Cite