1 V-topologies, t-Henselian Fields and Definable Valuations Katharina Dupont Department of Mathematics University of Constance British Postgraduate Model Theorie Conference, 2011
2 Outline V-Topologies and t-Henselian Fields 1 V-Topologies Local Sentences t-Henselian Fields Definable Valuations 2 p-adic Valuations Definable Valuations on t-Henselian Fields Real Closed Fields
3 V-Topology (Definition) Definition and Theorem Let K a field and B ⊆ P ( K ) such that � B := � U ∈B U = { 0 } and { 0 } / ∈ B 1 ∀ U , V ∈ B ∃ W ∈ B W ⊆ U ∩ V 2 ∀ U ∈ B ∃ V ∈ B V − V ⊆ U 3 ∀ U ∈ B ∀ x , y ∈ K ∃ V ∈ B ( x + V ) ( y + V ) ⊆ xy + U 4 ( x + V ) − 1 ⊆ x − 1 + U ∀ x ∈ K × ∀ U ∈ B ∃ V ∈ B 5 ∀ U ∈ B ∃ V ∈ B ∀ x , y ∈ K xy ∈ V ⇒ x ∈ U ∨ y ∈ U 6 T B := { U ⊆ K | ∀ x ∈ U ∃ V ∈ B x + V ⊆ U } is a topology on K . Such a topology is called V-topology .
4 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 or a valuation on K whose induced topology coincides with T .
5 Terms Definition We define terms as follows The constants 0 , 1 are terms. 1 The variables x , y are terms. 2 If t 1 and t 2 are terms so are t 1 + t 2 , t 1 − t 2 und t 1 · t 2 . 3
6 Formulas Definition We define formulas as follows If t 1 and t 2 are terms and U is a set variable then t 1 ˙ = t 2 and 1 t 1 ∈ U are formulas. We will call this formulas prime formulas . If ϕ, ψ are formulas, x is a variable and U is a set variable 2 then ¬ ϕ , ϕ ∧ ψ , ϕ ∨ ψ , ∃ x ϕ , ∀ x ϕ , ∃ U ϕ und ∀ U ϕ are formulas.
7 Sentences Definition A sentence is a formula without free variables and set variables.
8 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.
9 Positive and Negative Formulas Definition A formula in negation normal form is positive in U if U does 1 not occur in any negated prime subformula. A formula in negation normal form is negative in U if U only 2 occurs in negated prime subformulas.
10 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 .
11 Examples of Local Sentences Example The sentences defining V-topologies are local sentences. For example the first one � � B := U = { 0 } and { 0 } / ∈ B U ∈B can be written in the form ∀ U 0 ∈ U ∧ ∀ x ( ∃ U ¬ x ∈ U ∨ x ˙ = 0 ) ∧ ∀ U ∃ x ( x ∈ U ∧ ¬ x ˙ = 0 )
12 Local Equivalence Definition Two filtered fields are called locally equivalent when the same local sentences are true in both fields.
13 t-Henselian Fields Definition A filtered field is called t-henselian if it is locally equivalent 1 to a filtered field for which the filter is defined by a henselian valuation. A V-topological field is called t-henselian if the field with the 2 filter of zero-neighborhoods is t-henselian.
14 Characterization of t-Henselian Fields Theorem For a V-topological field ( K , T ) is equivalent: ( K , T ) is t-henselian. 1 For every n ≥ 2 exists an open zero neighborhood U such 2 that every polynominal X n + X n − 1 + a n − 2 X n − 2 + · · · + a 0 ∈ K [ X ] with a i ∈ U ( 0 ≤ i ≤ n − 2 ) has a zero in K. For every n ∈ N is the set of all polynominals 3 X n + a n − 1 X n − 1 + · · · + a 0 ∈ K [ X ] which have a simple zero in K open.
15 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 O v = { x ∈ K | ϕ ( x ) } .
16 Example: p-Adic Valuations Example For any prime number p the p -adic valuation on Q p is definable. It is Z p = { x ∈ Q p | ∃ y y 2 − y = px 2 } .
17 proof: Z p ⊇ { x ∈ Q p | ∃ y y 2 − y = px 2 } Proof: Let v denote the p -adic valuation for some prime p . Let x , y ∈ K such that y 2 − y = px 2 . Suppose x / ∈ Z p . 1.case v ( y ) < 0: Then v ( y 2 − y ) = min � � v ( y 2 ) , v ( y ) = 2 v ( y ) is even. But v ( px 2 ) = v ( p ) + v ( x 2 ) = 1 + 2 v ( x ) is uneven. This contradicts y 2 − y = px 2 . �
18 proof: Z p ⊇ { x ∈ Q p | ∃ y y 2 − y = px 2 } Proof: 2.case v ( y ) ≥ 0: Then v ( y 2 − y ) ≥ min � � v ( y 2 ) , v ( y ) ≥ 0 . But as v ( x ) < 0 and the value group of v is Z it is v ( x ) ≤ − 1 and therefore v ( px 2 ) = v ( p ) + 2 v ( x ) = 1 + 2 v ( x ) ≤ 1 − 2 = − 1 < 0 . This again contradicts y 2 − y = px 2 . �
19 proof: Z p ⊆ { x ∈ Q p | ∃ y y 2 − y = px 2 } Proof: On the other hand if x ∈ Z p it is ′ = 2 T − 1 and f := T 2 − T − px 2 ∈ Z p [ T ] with f = T 2 − T and f therefore 0 is a simple root of f . By Hensel’s Lemma f has a simple root y ∈ Z p and for this y y 2 − y = px 2 holds. �
20 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. O v = { x ∈ K | ∃ π ( ∃ y y 2 − y = π x 2 ∧∃ z ∀ y ¬ y 2 − y = π z 2 ∧∀ a ∀ b ( ∃ y y 2 − y = π a 2 b 2 ∨∀ y ¬ y 2 − y = π a 2 ∨∀ y ¬ y 2 − y = π b 2 )) } .
21 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?
22 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 .
23 real closed fields Theorem On a real closed field only the trivial valuation is definable.
24 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 p 1 ˙ = p 2 and p 1 ≤ p 2 for polynominals p 1 , p 2 ∈ K [ X ] where without loss of generality we can assume p 2 = 0. Both kinds of formulas define sets which are finite unions of intervals. �
25 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. �
26 Proof: On Real Closed Fields Only The Trivial Valuation Is Definable (3) Proof: Let O a valuation ring on K with O = � 1 ≤ i ≤ n � a i , b i � for some a i ∈ K ∪ {−∞} and b i ∈ 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 � a j , b j � ∩ N infinite and therefore b j = ∞ . Suppose there exists x ∈ K with x / ∈ O . There exists an m ∈ N with m + x > a j and therefore m + x ∈ � a j , ∞ ) = � a j , b j � ⊂ O . But this is a contradiction as v ( m + x ) = min { v ( x ) , v ( m ) } = v ( x ) < 0. Therefore O is trivial. �
27 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. �
28 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
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