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On elementary equivalence, isomorphism and isogeny

Pete L. Clark (2006)

Journal de Théorie des Nombres de Bordeaux

Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero...

On Galois cohomology and realizability of 2-groups as Galois groups II

Ivo Michailov (2011)

Open Mathematics

In [Michailov I.M., On Galois cohomology and realizability of 2-groups as Galois groups, Cent. Eur. J. Math., 2011, 9(2), 403–419] we calculated the obstructions to the realizability as Galois groups of 14 non-abelian groups of order 2n, n ≥ 4, having a cyclic subgroup of order 2n−2, over fields containing a primitive 2n−3th root of unity. In the present paper we obtain necessary and sufficient conditions for the realizability of the remaining 8 groups that are not direct products of smaller groups....

On Galois cohomology and realizability of 2-groups as Galois groups

Ivo Michailov (2011)

Open Mathematics

In this paper we develop some new theoretical criteria for the realizability of p-groups as Galois groups over arbitrary fields. We provide necessary and sufficient conditions for the realizability of 14 of the 22 non-abelian 2-groups having a cyclic subgroup of index 4 that are not direct products of groups.

On generalized “ham sandwich” theorems

Marek Golasiński (2006)

Archivum Mathematicum

In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let A 1 , ... , A m ⊆ ℝ n be subsets with finite Lebesgue measure. Then, for any sequence f 0 , ... , f m of ℝ -linearly independent polynomials in the polynomial ring ℝ [ X 1 , ... , X n ] there are real numbers λ 0 , ... , λ m , not all zero, such that the real affine variety { x ∈ ℝ n ; λ 0 f 0 ( x ) + ⋯ + λ m f m ( x ) = 0 } simultaneously bisects each of subsets A k , k = 1 , ... , m . Then some its applications are studied.

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