Narrow Results

The aim of this paper is to provide a proof of the following result claimed by Albu (Infinite field extensions with Galois–Cogalois correspondence (II), Revue Roumaine Math. Pures Appl. 47 (2002), to appear): The Kneser group Kne(E/F) of an Abelian G-Cogalois extension E/F and the group of continuous characters Ch(Gal(E/F)) of its Galois group Gal(E/F) are isomorphic (in a noncanonical way). The proof we give in this paper explains why such an isomorphism is expected, being based on a classical result of Baer (Amer. J. Math.61 (1939), 1–44) devoted to the existence of group isomorphisms arising from lattice isomorphisms of their lattices of subgroups.

Let p be a prime number and let ℚ/ℤ′ be the elements in ℚ/ℤ of order prime to p. Let , where c is a prime power of p. We use characters and valuation theory to prove that Δ is a parameter space for the cyclic tame extensions of the formal Laurent series field k_p((t)) of degree prime to p. Furthermore, we construct the cyclic tame extension corresponding to a given triple in Δ. The structure of finite cyclic tame extensions of the p-adic number fields was thoroughly investigated by A. A. Albert in 1935. Here we get the same result as consequence of our main theorem.

This paper aims to describe irreducible restricted modules of the special Hamiltonian Lie superalgebras of odd type over an algebraically closed field of characteristic $p>3$. A sufficient and necessary condition for the restricted Kac modules to be irreducible is given in terms of typical weights. Furthermore, the character formulas for the irreducible quotients of the restricted Kac modules are reduced to the ones for the irreducible quotients of the restricted Kac modules of the Hamiltonian Lie superalgebras of odd type and the ones of a $1$-dimensional central extension of the classical Lie superalgebra of type $\widetilde{\mathrm{\text{P}}}(n)$. In particular, the composition factors of restricted Kac modules are determined in a sense.