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Note on a variation of the Schröder-Bernstein problem for fields

F. S. Cater (2002)

Czechoslovak Mathematical Journal

In this note we study fields F with the property that the simple transcendental extension F ( u ) of F is isomorphic to some subfield of F but not isomorphic to F . Such a field provides one type of solution of the Schröder-Bernstein problem for fields.

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 ruled fields

Jack Ohm (1989)

Journal de théorie des nombres de Bordeaux

Some results and problems that arise in connection with the foundations of the theory of ruled and rational field extensions are discussed.

On subfields of k ( x )

Victor Alexandru, Nicolae Popescu (1986)

Rendiconti del Seminario Matematico della Università di Padova

On the dimension of the space of ℝ-places of certain rational function fields

Taras Banakh, Yaroslav Kholyavka, Oles Potyatynyk, Michał Machura, Katarzyna Kuhlmann (2014)

Open Mathematics

We prove that for every n ∈ ℕ the space M(K(x 1, …, x n) of ℝ-places of the field K(x 1, …, x n) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1, …, x n)) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dimℤ M(K(x 1, x 2)) = 2 and the cohomological dimension dimG M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G.

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