Finite quotients of Galois pro- $$p$$ p groups and rigid fields

Abstract

For a prime number p, the author shows that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-p group G coincide, then G has a very simple structure, i.e., G is a p-adic analytic pro-p group (see Theorem 1). This result has a remarkable Galoistheoretic consequence: if the two corresponding canonical finite extensions of a field F-with F containing a primitive p-th root of unity-coincide, then F is p-rigid (see Corollary 1). The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-p groups.Résumé Étant donné un nombre premier p, l'auteur montre que si G est un pro-p groupe de Bloch-Kato de type fini dont deux quotients finis canoniques coïncident, alors G admet une structure est très simple: c'est un pro-p groupe p-adique analytique (voir théorème 1). Ce résultat a une conséquence remarquable pour la théorie de Galois: si un corps F contient une racine primitive p-ième de l'unité et que ses deux extensions canoniques finies correspondantes coïncident, F est alors p-rigide (voir corollaire 1). La démonstration s'appuie sur des résultats de théorie des groupes et exploite certaines propriétés des pro-p groupes de Bloch-Kato.

Rapid Photoactivated Generation of Nitroxyl (HNO) under Neutral pH Conditions

“…[14] To date,alimited number of photocontrollable HNO donors have been reported based on acyl nitroso intermediate formation by aretro-Diels-Alder reaction [9a-d] or the retrocycloaddition of a1 ,2,4-oxadiazole-4-oxide. [9a,e,f] Limitations of these HNO donors include photodecomposition in ambient light, [15] and/ or competition from NO formation, [9b-d] background thermal reactions,a nd secondary photocleavage processes. [16] Herein, we present the first in an ew family of photoactivatable N-alkoxysulfonamide HNO donors, 1,incorporating the well-characterized O-(3-hydroxy-2-naphthalenyl)methyl (HNM) photolabile protecting group.T he HNM group was selected as the phototrigger because of its rapid photocleavage (k release % 10 5 s À1 (t 1/2 % ms)) with good quantum and chemical yields.…”

From Cyclopentadiene to Isoxazoline‐Carbocyclic Nucleosides; Synthesis of Highly Active Inhibitors of Influenza A Virus H1N1

“…A crowded group of signals between δ = 7.5-8.5 ppm were attributed to the quinoline ring, whereas the bridged methylene gave two doublets at δ = 1.95 and 2.28 ppm. The yield of cycloadduct 14 are somewhat lower than usual for this kind of nitrosocarbonyl cycloaddition reactions, [8] and we could identify the regioisomeric 1,3-dipolar cycloadducts 15a and 15b as by-products, which were isolated in 23 and 18 % yields, respectively. The structures of cycloadducts 15a and 15b were determined on the basis of their analytical and spectroscopic data.…”

“…By using the latter approach, Keck showed that the nitrosocarbonyl intermediates, thermally liberated from their HDA adducts with 9,10-dimethylanthracene in benzene at reflux, behave as active enophiles with various olefins in an ene-addition to afford adducts 4. [7] Recently, we developed two alternative routes to nitrosocarbonyl compounds 1 through the mild oxidation of nitrile oxides 5 with N-methylmorpholine N-oxide (NMO) [8] and the clean photolysis and thermolysis of 1,2,4oxadiazole-4-oxides 6. [9] The mild oxidation of nitrile oxide with NMO complements the HDA route and the ene-route, which works well with tetra-and tri-substituted alkenes.…”