Algebraic patching over complete domains

2009

Ann. Math.

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We prove that every finite split embedding problem is solvable over the field K..X 1 ; : : : ; X n // of formal power series in n 2 variables over an arbitrary field K, as well as over the field Quot.AOEOEX 1 ; : : : ; X n / of formal power series in n 1 variables over a Noetherian integrally closed domain A. This generalizes a theorem of Harbater and Stevenson, who settled the case K..X 1 ; X 2 //.

Section: Rings Of Convergent Power Seriesmentioning

confidence: 63%

Section: Solution Of Split Embedding Problemsmentioning

confidence: 81%

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Split embedding problems over complete domains

2009

Ann. Math.

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“…Proof The existence of a is given in the proof of [14,Lemma 6.3(b)]. Suppose β is then the corresponding primitive element for θ a (F)/E.…”

Section: Lemma 32 Let F Be a Galois Extension Of E Contained In K ((W)mentioning

confidence: 98%

“…Indeed, by [14,Lemma 6.4(b)] (applied with D =K , c = α, =Q k , and p = w 1 − r c−c 1 there replaced with p j = w j − r α−c j here) ϕ mapsR I homomorphically intoK (by substitution of each w i with r α−c i ) and the kernel of this homomorphism is generated by the prime element p j . Moreover, the valuation ring of ϕ inQ k is the localization ofR k with respect to p jRk .…”

Section: Proposition 13 Fix K ∈ I and Letmentioning

confidence: 99%

Galois theory over complete local domains

2010

Math. Ann.

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Complete local domains play an important role in commutative algebra and algebraic geometry, and their algebraic properties were already described by Cohen's structure theorem in 1946. However, the Galois theoretic properties of their quotient fields only recently began to unfold. In 2005 Harbater and Stevenson considered the two dimensional case. They proved that the absolute Galois group of the field K ((X, Y )) (where K is an arbitrary field) is semi-free. In this work we settle the general case, and prove that if R is a complete local domain of dimension exceeding 1, then the quotient field of R has a semi-free absolute Galois group. IntroductionLet K be an arbitrary field. Recall [6, Sect. 2] that if every non-trivial finite split embedding problem over K has |K | solutions, one says that the absolute Galois group of K is quasi-free. In a recent work [1] the stronger notion of semi-free profinite groups was introduced. The absolute Galois group of K is semi-free if each non-trivial finite split embedding problem has |K | independent solutions. More explicitly, Gal(K ) is semi free, if for any finite Galois extension L/K with Galois group acting on a finite group G, there exists a family of fields {L i } i∈I of cardinality |I | = |K |, such that each L i is a Galois extension of K with group G , L i contains L, and the projection G → is the restriction map Gal(L i /K ) → Gal(L/K ). Moreover the fields {L i } i∈I are linearly disjoint over L.

Split embedding problems over the open arithmetic disc

Fehm

Trans. Amer. Math. Soc.

2014

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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.