Linear theories of global fields with absolute values
Abstract
We study the theory of a global field k as a k-vector space with a predicate for one of the absolute values on k. For example, we prove that in this language a global field with an ultrametric or real archimedean absolute value has a decidable theory, while with a complex absolute value the theory is always undecidable. We also study the existential theories and axiomatize k together with predicates for all non-complex absolute values on k simultaneously.
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