Value groups, residue fields, and bad places of rational function fields

DOI10.1090/S0002-9947-04-03463-4zbMath1122.12005OpenAlexW1538184085MaRDI QIDQ4813862

Franz-Viktor Kuhlmann

Publication date: 13 August 2004

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0002-9947-04-03463-4

zbMATH Keywords

extensions of valuations to rational function fieldsconstruction of extensions with prescribed value groups and residue fields

Mathematics Subject Classification ID

Valuations, completions, formal power series and related constructions (associative rings and algebras) (16W60) Ordered fields (12J15) Valued fields (12J10)

Related Items (57)

Kronecker Function Rings of Transcendental Field Extensions ⋮ Places of algebraic function fields in arbitrary characteristic ⋮ Henselianity in the language of rings ⋮ Counting the number of distinct distances of elements in valued field extensions ⋮ Prüfer intersection of valuation domains of a field of rational functions ⋮ The algebra and model theory of tame valued fields ⋮ Embedding Henselian fields into power series ⋮ Cuts and small extensions of abelian ordered groups ⋮ On the ranks and implicit constant fields of valuations induced by pseudo monotone sequences ⋮ NOTES ON EXTREMAL AND TAME VALUED FIELDS ⋮ Limit groups and groups acting freely on \(\mathbb{R}^n\)-trees. ⋮ Subfields of algebraically maximal Kaplansky fields ⋮ Rigidity of valuative trees under Henselization ⋮ Valuation-transcendental extensions and pseudo-monotone sequences ⋮ On the irreducible factors of a polynomial. II. ⋮ Approximation types describing extensions of valuations to rational function fields ⋮ Minimal pairs, inertia degrees, ramification degrees and implicit constant fields ⋮ Extensions of valuations to rational function fields over completions ⋮ Valuations and orderings on the real Weyl algebra ⋮ Distances of elements in valued field extensions ⋮ Transcendental extensions of a valuation domain of rank one ⋮ Density of composite places in function fields and applications to real holomorphy rings ⋮ A ruled residue theorem for algebraic function fields of curves of prime degree ⋮ Defect extensions and a characterization of tame fields ⋮ The defect formula ⋮ Characterizing NIP henselian fields ⋮ Tame key polynomials ⋮ Geometric parametrization of valuations on a polynomial ring in one variable ⋮ Abstract key polynomials and distinguished pairs ⋮ MacLane-Vaquié chains and valuation-transcendental extensions ⋮ Valuations with an infinite limit-depth ⋮ Key polynomials and minimal pairs ⋮ A classification of Artin-Schreier defect extensions and characterizations of defectless fields ⋮ The Zariski–Riemann space of valuation domains associated to pseudo-convergent sequences ⋮ A construction for a class of valuations of the field \(k(X_1,\cdots ,X_d,Y)\) with large value group ⋮ Minimal pairs, minimal fields and implicit constant fields ⋮ Algebraic independence of elements in immediate extensions of valued fields ⋮ Affine schemes and topological closures in the Zariski-Riemann space of valuation rings ⋮ Ramification of Valuations and Local Rings in Positive Characteristic ⋮ Elimination of ramification. II: Henselian rationality ⋮ Infinite towers of Galois defect extensions of Kaplansky fields ⋮ On common extensions of valued fields ⋮ Valuations on rational function fields that are invariant under permutation of the variables ⋮ Extending valuations to the field of rational functions using pseudo-monotone sequences ⋮ Every place admits local uniformization in a finite extension of the function field ⋮ MacLane-Vaquié chains of valuations on a polynomial ring ⋮ A CONJECTURAL CLASSIFICATION OF STRONGLY DEPENDENT FIELDS ⋮ On truncations of valuations ⋮ Extensions of a valuation from $K$ to $K[x$] ⋮ Realizations of semilocal ℓ-groups over k [ x ₁ , x ₂ ,… x _n ] ⋮ ALL NON-ARCHIMEDEAN NORMS ON K[X₁, . . ., X_r] ⋮ Corrections and notes to “Value groups, residue fields and bad places of rational function fields” ⋮ Eliminating tame ramification generalizations of Abhyankar's lemma ⋮ A VALUATION THEORETIC CHARACTERIZATION OF RECURSIVELY SATURATED REAL CLOSED FIELDS ⋮ Valuative trees over valued fields ⋮ On the implicit constant fields and key polynomials for valuation algebraic extensions ⋮ Counterexamples to local monomialization in positive characteristic

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