On common extensions of valued fields
نویسندگان
چکیده
Given a valuation v on field K, an extension v¯ to algebraic closure K¯ and w K(X). In this paper, we study common extensions w¯ of both the K¯(X). We describe set valuations using last key polynomial for its roots.
منابع مشابه
Maximal Immediate Extensions of Valued Differential Fields
We show that every valued differential field has an immediate strict extension that is spherically complete. We also discuss the issue of uniqueness up to isomorphism of such an extension.
متن کاملBasis discrepancies for extensions of valued fields
Let F be a field complete for a real valuation. It is a standard result in valuation theory that a finite extension of F admits a valuation basis if and only if it is without defect. We show that even otherwise, one can construct bases in which the discrepancy between measuring valuation an element versus on the components in its basis decomposition can be made arbitrarily small. The key step i...
متن کاملCommon extensions of semigroup- valued charges
Let A and B be fields of subsets of a nonempty set X and let μ : A → E and ν : B → E be finitely additive measures (“charges”) taking values in a commutative semigroup E. We assume that μ and ν are consistent (e.g. μ = ν on A ∩ B) and ask whether they have a common extension to a charge ρ : A ∨ B → E. Now, we shall see (proposition 2.3) that the most natural consistency condition which we can f...
متن کاملAn Isomorphism Theorem for Henselian Algebraic Extensions of Valued Fields
In general, the value groups and the residue elds do not suuce to classify the algebraic henselian extensions of a valued eld K, up to isomorphism over K. We deene a stronger, yet natural structure which carries information about additive and multiplica-tive congruences in the valued eld, extending the information carried by value groups and residue elds. We discuss the cases where these \mixed...
متن کاملOn Pseudo Algebraically Closed Extensions of Fields
The notion of ‘Pseudo Algebraically Closed (PAC) extensions’ is a generalization of the classical notion of PAC fields. In this work we develop a basic machinery to study PAC extensions. This machinery is based on a generalization of embedding problems to field extensions. The main goal is to prove that the Galois closure of any proper separable algebraic PAC extension is its separable closure....
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Algebra
سال: 2021
ISSN: ['1090-266X', '0021-8693']
DOI: https://doi.org/10.1016/j.jalgebra.2021.04.027