Elad Paran

Fully Hilbertian fields

We introduce the notion of fully Hilbertian fields, a strictly stronger notion than that of Hilbertian fields. We show that this class of fields exhibits the same good behavior as Hilbertian fields, but for fields of uncountable cardinality, is more natural than the notion of Hilbertian fields. In particular, we show it can be used to achieve stronger Galois theoretic results. Our proofs also provide a step toward the so-called Jarden-Lubotzky twinning principle.

Klein Approximation and Hilbertian fields

Abstract The quotient field of a generalized Krull domain of dimension exceeding one is Hilbertian by a theorem of Weissauer. Building on work of R. Klein we generalize this criterion for Hilbertianity to a wider class of domains. This allows us to extend recent results on Hilbertianity of fields of power series and obtain new Hilbertian fields.

Hilbertianity of fields of power series

Let R be a domain contained in a rank-1 valuation ring of its quotient field. Let R[[X]] be the ring of formal power series over R, and let F be the quotient field of R[[X]]. We prove that F is Hilbertian. This resolves and generalizes an open problem of Jarden, and allows to generalize previous Galois-theoretic results over fields of power series. MR Classification: 12E30 Introduction Field Arithmetic studies the connection between the arithmetic properties of a field and its Galois theoretic properties. A central conjecture in Field Arithmetic, which widely generalizes the inverse Galois problem, was coined by Debes and Deschamps: Conjecture A [DD97, §2.1.2]: If F is a Hilbertian field, then every finite split embedding problem over F is solvable. Conjecture A is proven in [Po96] in the case where F is ample (called “large” in [Po96]). In particular, this holds if F is complete with respect to a discrete valuation. The author was supported by an ERC grant while working on this research. Following this came a series of works studying Galois theory over complete valued domains whose quotient fields are not complete. The archetype of such a domain is A[[X]], where A is some domain which is not a field. The first Galois theoretic result over such fields is due to Lefcourt [Le99], who showed that if A is integrally closed and Noetherian, then the inverse Galois problem has a positive solution over F = Quot(A[[X]]). We call the field F the field of formal power series over A (note that F is usually smaller than the field K((X)) = Quot(K[[X]]) of formal power series over K = Quot(A)). The next result is due to Harbater-Stevenson [HaS05], who showed that Conjecture A holds over F , in the case where A is a complete discrete valuation ring (moreover, they showed that each such problem has |F |-many distinct solutions). Then the author [Pa09] showed that Conjecture A holds in a more general situation: Theorem B: Let A be a Krull domain (e.g. an integrally closed Noetherian domain). Then every finite split embedding problem over Quot(A[[X]]) is solvable. In [Po09], Pop showed that the quotient field of a Henselian domain is ample. Using his result, one can give a short proof of Theorem B – given a split embedding problem over F , extend it to F (t), solve it there using the result of [Po96], then specialize the solution into a solution over F . The second part of the proof, specialization, is possible since F is Hilbertian by a theorem of Weissauer [Ws82]. That theorem asserts that the quotient field of a domain R of dimension exceeding 1 is Hilbertian, provided that R is a generalized Krull domain. We recall the definition: Definition C: A domain R is called a generalized Krull domain if there exists a non-empty family F of non-trivial rank-1 valuations on K = Quot(R), satisfying the following properties: (a) Denoting the valuation ring of v by Rv for each v ∈ F , we have ⋂ v∈F Rv = R. (b) For each a ∈ K×, v(a) = 0 for all but finitely many v ∈ F . (c) For each v ∈ F , Rv is the localization of R with respect to the center p(v) = {a ∈ R | v(a) > 0} of v on R. If every v ∈ F is discrete, then R is called a Krull domain [ZaS60, §VI.13]. Note that by assuming that F is non-empty, we do not consider fields as Krull domains.

Split embedding problems over function fields over complete domains

Let

On Totally Ramified Extensions of Discrete Valued Fields

R

Split embedding problems over complete domains

be a Krull domain, complete with respect to a nonzero ideal. Let

Algebraic patching over complete domains

K

Power series over generalized Krull domains

be the quotient field of~

Patching and admissibility over two-dimensional complete local domains

R

Galois theory over complete local domains

. We prove that every finite split embedding problem is solvable over every function field in one variable over~