Regular Split Embedding Problems over Function Fields of One Variable over Ample Fields

Haran, Dan; Jarden, Moshe

doi:10.1006/jabr.1998.7454

Cited by 22 publications

(12 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It can also be seen by choosing a parameter x for a point P on the closed fibre X ofX, constructing the building blocks for the x-line over T , and then taking a base change to the local ring at P (which, beingétale, preserves total ramification). This contrasts with the strategy in [7], Proposition 1.4, which is to map a curve to the line, perform a patching construction there, and then deduce a result about the curve. (b) Alternatively, the above proof can be extended to more general smooth curves over T by using Theorem 5.10 instead of Theorem 4.14 (where the complete local ring is independent of which smooth curve is taken).…”

Section: Remark 74mentioning

confidence: 65%

Patching over fields

2010

View full text Add to dashboard Cite

We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and vector spaces, rather than rings and modules. After presenting a self-contained development of this form of patching, we obtain applications to other structures such as Brauer groups and differential modules.

show abstract

Section: Remark 74mentioning

confidence: 65%

Patching over fields

2010

View full text Add to dashboard Cite

show abstract

“…We refer the reader to [HJ3,Thm. B] for an exact formulation of a split embedding problem and for an algebraic proof of the statement.…”

Section: The Main Resultsmentioning

confidence: 99%

PSC Galois Extensions of Hilbertian Fields

Geyer¹,

Jarden

2002

Math. Nachr.

Self Cite

View full text Add to dashboard Cite

We prove the following result: Theorem. Let K be a countable Hilbertian field, S a finite set of local primes of K, and e ≥ 0 an integer. Then, for almost all σ ∈ G (K)e, the field Ks [σ] ∩ Ktot,S is PSC. Here a local prime is an equivalent class 𝔭 of absolute values of K whose completion is a local field, $ \hat K $𝔭. Then K𝔭 = Ks ∩ $ \hat K $𝔭 and Ktot,S = ∩𝔭 ∈ S∩σ ∈ G(K) Kσ𝔭. G(K) stands for the absolute Galois group of K. For each σ = (σ1, …, σe ) ∈ G(K)e we denote the fixed field of σ1, …, σe in Ks by Ks(σ). The maximal Galois extension of K in Ks(σ) is Ks[σ]. Finally “almost all” means “for all but a set of Haar measure zero”.

show abstract

“…In [DD99] it is conjectured that every FSEP over a Hilbertian field is solvable. This has been proven for function fields over ample fields in [Pop96,HJ98b]. The field E above can be viewed as an analogue of the (non-ample) field K((x))(t) of rational functions over the complete (and hence ample) field K((x)) (where K is some field), see discussion in [Har88,p.…”

Section: Part a Enlarging Kmentioning

confidence: 91%

Split embedding problems over the open arithmetic disc

Fehm

Paran

2014

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Let Z{t} be the ring of arithmetic power series that converge on the complex open unit disc. A classical result of Harbater asserts that every finite group occurs as a Galois group over the quotient field of Z{t}. We strengthen this by showing that every finite split embedding problem over Q acquires a solution over this field. More generally, we solve all t-unramified finite split embedding problems over the quotient field of O K {t}, where O K is the ring of integers of an arbitrary number field K.

show abstract

Regular Split Embedding Problems over Function Fields of One Variable over Ample Fields

Abstract: * Pop, who introduces this type of fields in [Po1], calls them 'large'. Since this name has been used earlier with a different meaning, we have modified it to 'ample' [HaJ, Definition 6.3 and the attached footenote].

Cited by 22 publications

References 6 publications

Patching over fields

Patching over fields

PSC Galois Extensions of Hilbertian Fields

Split embedding problems over the open arithmetic disc

Contact Info

Product

Resources

About