An Isomorphism Theorem for Henselian Algebraic Extensions of Valued Fields
نویسنده
چکیده
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 structures" give a solution of the classiication problem.
منابع مشابه
Automorphisms and Isomorphisms of Real Henselian Fields
Let K and L be ordered algebraic extensions of an ordered field F. Suppose K and L are Henselian with Archimedean real closed residue class fields. Then K and L are shown to be F-isomorphic as ordered fields if they have the same value group. Analogues to this result are proved involving orderings of higher level, unordered extensions, and, when K and L are maximal valued fields, transcendental...
متن کاملSpringer’s Theorem for Tame Quadratic Forms over Henselian Fields
A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tamely ramified extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation, hence also isomorphic to a direct sum of copies of...
متن کاملA p-ADIC EXAMPLE FOR THE CHARACTERIZATION OF THE CANONICAL p-HENSELIAN VALUATION
The authors have shown recently that the canonical p-henselian valuation is uniformly ∅-definable in the elementary class of fields which have characteristic p or contain a primitive pth root of unity ζp. In order to do this, we proved a classification of the canonical p-henselian valuation via case distinction. One of the cases discussed was previously not even known for algebraic extensions o...
متن کاملThe Defect
We give an introduction to the valuation theoretical phenomenon of “defect”, also known as “ramification deficiency”. We describe the role it plays in deep open problems in positive characteristic: local uniformization (the local form of resolution of singularities), the model theory of valued fields, the structure theory of valued function fields. We give several examples of algebraic extensio...
متن کاملThe Algebra and Model Theory of Tame Valued Fields
A henselian valued field K is called a tame field if its algebraic closure K̃ is a tame extension, that is, the ramification field of the normal extension K̃|K is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax–Kochen– Ershov Principles for tame fields. This leads to model compl...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1992