V-topologies, t-Henselian Fields and Definable Valuations Katharina - - PowerPoint PPT Presentation

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V-topologies, t-Henselian Fields and Definable Valuations

Katharina Dupont

Department of Mathematics University of Constance

British Postgraduate Model Theorie Conference, 2011

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Outline

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V-Topologies and t-Henselian Fields V-Topologies Local Sentences t-Henselian Fields

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Definable Valuations p-adic Valuations Definable Valuations on t-Henselian Fields Real Closed Fields

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V-Topology (Definition)

Definition and Theorem Let K a field and B ⊆ P (K) such that

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B :=

U∈B U = {0} and {0} /

∈ B

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∀ U, V ∈ B ∃ W ∈ B W ⊆ U ∩ V

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∀ U ∈ B ∃ V ∈ B V − V ⊆ U

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∀ U ∈ B ∀ x, y ∈ K ∃ V ∈ B (x + V) (y + V) ⊆ xy + U

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∀ U ∈ B ∀ x ∈ K × ∃ V ∈ B (x + V)−1 ⊆ x−1 + U

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∀ U ∈ B ∃ V ∈ B ∀ x, y ∈ K xy ∈ V ⇒ x ∈ U ∨ y ∈ U TB := {U ⊆ K | ∀ x ∈ U ∃ V ∈ B x + V ⊆ U} is a topology on K. Such a topology is called V-topology.

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V-Topology (Theorem)

Theorem Let K be a field and T a topology on K. Then T is a V-topology if and only if there exists either an archimedean absolute value

• r a valuation on K whose induced topology coincides with T .

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Terms

Definition We define terms as follows

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The constants 0, 1 are terms.

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The variables x, y are terms.

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If t1 and t2 are terms so are t1 + t2, t1 − t2 und t1 · t2.

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Formulas

Definition We define formulas as follows

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If t1 and t2 are terms and U is a set variable then t1 ˙ =t2 and t1 ∈ U are formulas. We will call this formulas prime formulas.

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If ϕ, ψ are formulas, x is a variable and U is a set variable then ¬ϕ, ϕ ∧ ψ, ϕ ∨ ψ, ∃ x ϕ, ∀ x ϕ, ∃ U ϕ und ∀ U ϕ are formulas.

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Sentences

Definition A sentence is a formula without free variables and set variables.

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Negation Normal Form

Definition A formula has negation normal form if ¬ occurs only in front of prime formulars. Theorem Each formula is equivalent to a formula in negation normal form.

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Positive and Negative Formulas

Definition

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A formula in negation normal form is positive in U if U does not occur in any negated prime subformula.

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A formula in negation normal form is negative in U if U only

• ccurs in negated prime subformulas.

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Local Formulas

Definition A formula is called local if the equivalent formula in negation normal form is build from prime formulas and negated prime formulas using ∧, ∨, ∃, ∀ such that: ∃ U only occurs in front of formulas which are negative in U. ∀ U only occurs in front of formulas which are positive in U.

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Examples of Local Sentences

Example The sentences defining V-topologies are local sentences. For example the first one

• B :=
• U∈B

U = {0} and {0} / ∈ B can be written in the form ∀ U 0 ∈ U ∧ ∀ x (∃ U ¬x ∈ U ∨ x ˙ =0) ∧ ∀ U ∃ x (x ∈ U ∧ ¬x ˙ =0)

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Local Equivalence

Definition Two filtered fields are called locally equivalent when the same local sentences are true in both fields.

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t-Henselian Fields

Definition

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A filtered field is called t-henselian if it is locally equivalent to a filtered field for which the filter is defined by a henselian valuation.

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A V-topological field is called t-henselian if the field with the filter of zero-neighborhoods is t-henselian.

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Characterization of t-Henselian Fields

Theorem For a V-topological field (K, T ) is equivalent:

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(K, T ) is t-henselian.

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For every n ≥ 2 exists an open zero neighborhood U such that every polynominal X n + X n−1 + an−2X n−2 + · · · + a0 ∈ K[X] with ai ∈ U (0 ≤ i ≤ n − 2) has a zero in K.

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For every n ∈ N is the set of all polynominals X n + an−1X n−1 + · · · + a0 ∈ K[X] which have a simple zero in K open.

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Definable Valuations (Definition)

Definition Let L = (0, 1; +, ·, −) the language of rings. We call a valuation v on a field K definable if there exists a L-formula ϕ in one variable such that Ov = {x ∈ K | ϕ(x)}.

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Example: p-Adic Valuations

Example For any prime number p the p-adic valuation on Qp is definable. It is Zp = {x ∈ Qp | ∃ y y2 − y = px2}.

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proof: Zp ⊇ {x ∈ Qp | ∃ y y2 − y = px2}

Proof: Let v denote the p-adic valuation for some prime p. Let x, y ∈ K such that y 2 − y = px2. Suppose x / ∈ Zp. 1.case v(y) < 0: Then v(y2 − y) = min

• v(y2), v(y)
• = 2v(y)

is even. But v(px2) = v(p) + v(x2) = 1 + 2v(x) is uneven. This contradicts y 2 − y = px2.

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proof: Zp ⊇ {x ∈ Qp | ∃ y y2 − y = px2}

Proof: 2.case v(y) ≥ 0: Then v(y2 − y) ≥ min

• v(y2), v(y)
• ≥ 0.

But as v(x) < 0 and the value group of v is Z it is v(x) ≤ −1 and therefore v(px2) = v(p) + 2v(x) = 1 + 2v(x) ≤ 1 − 2 = −1 < 0. This again contradicts y 2 − y = px2.

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proof: Zp ⊆ {x ∈ Qp | ∃ y y2 − y = px2}

Proof: On the other hand if x ∈ Zp it is f := T 2 − T − px2 ∈ Zp[T] with f = T 2 − T and f

′ = 2T − 1 and

therefore 0 is a simple root of f. By Hensel’s Lemma f has a simple root y ∈ Zp and for this y y2 − y = px2 holds.

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Comments

Remark We only use that v is a henselian valuation with value group Z and v(p) = 1. Therefore the same proof works to show that for any field K on the field of formal powerseries K((T)) over K the valuation ring K[[T]] is definable. We can get rid of the parameters. Ov = {x ∈ K | ∃π (∃y y2 − y = πx2 ∧∃z ∀y ¬y2 − y = πz2 ∧∀a ∀b (∃y y2 − y = πa2b2 ∨∀y ¬y2 − y = πa2 ∨∀y ¬y2 − y = πb2))}.

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Leading Questions For Our Research

Question What conditions are sufficient for a field to admit a nontrivial definable valuation? What conditions are necessary for a field to admit a nontrivial definable valuation? What do we know about this definable valuation, if it exists?

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Definable Valuations on t-Henselian Fields

Theorem (Koenigsmann) Let (K, T ) a t-henselian field. If K is neither real closed nor separably closed then K admits a definable valuation inducing T .

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real closed fields

Theorem On a real closed field only the trivial valuation is definable.

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Proof: On Real Closed Fields Only The Trivial Valuation Is Definable

Proof: Let K an archimedean ordered real closed field. As real closed fields allow quantifier elimination in the language L≤ = (0, 1; +, −, ·; ≤) every valuation which is definable by an L-formula is definable by a quantifierfree L≤-formula. Terms in L≤ in one variable are polynominals therefore prime formulas in L≤ are of the form p1 ˙ =p2 and p1 ≤ p2 for polynominals p1, p2 ∈ K[X] where without loss of generality we can assume p2 = 0. Both kinds of formulas define sets which are finite unions of intervals.

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Proof: On Real Closed Fields Only The Trivial Valuation Is Definable (2)

Proof: If we take the compliment of a finite union of intervals or the intersection or union of two finite unions of intervals we get again a finite union of intervals. Therefore if ϕ and ψ define a finite union of intervals so do ¬ϕ, ϕ ∧ ψ and ϕ ∨ ψ. Therefore every set which is defined by a quantifierfree L≤-formula is a finite union of intervals.

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Proof: On Real Closed Fields Only The Trivial Valuation Is Definable (3)

Proof: Let O a valuation ring on K with O =

1≤i≤nai, bi for

some ai ∈ K ∪ {−∞} and bi ∈ K ∪ {∞} where denotes either ( or [ and denotes ) or ]. As O is a ring we have N ⊂ O and therefore there exits 1 ≤ j ≤ n with aj, bj ∩ N infinite and therefore bj = ∞. Suppose there exists x ∈ K with x / ∈ O. There exists an m ∈ N with m + x > aj and therefore m + x ∈ aj, ∞) = aj, bj ⊂ O. But this is a contradiction as v(m + x) = min{v(x), v(m)} = v(x) < 0. Therefore O is trivial.

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Proof: On Real Closed Fields Only The Trivial Valuation Is Definable (4)

Proof: The case of arbitrary real closed fields now follows from the fact that the theory of real closed fields is complete.

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• H. Duerbaum, H.-J. Kowalski, Arithmetische

Kennzeichnung von Koerpertopologien, J. reine angew.

• Math. 191 (1953), Seite 135-152

A.J. Engler, A. Prestel, Valued Fields, Springer Verlag, (2005)

• A. Prestel, M. Ziegler, Model theoretic methods in the

theory of topological fields, J. reine angew. Math. 299/300 (1978), Seite 318-341

• J. Koenigsmann, Definable Valuations