variants of hensel's lemma - hhureh.math.uni-duesseldorf.de/~severin/docs/hensel-gesamt... · 2017....

Variants of Hensel’s lemma

Florian Severin

Heinrich-Heine Universität Düsseldorf

GeSAMT III, 14 July 2017

Valuations

DefinitionA valuation on a field K is a surjective map v : K → vK ∪ {∞}, where(vK ,+,

Valuation rings

DefinitionA valuation ring on a field K is a subring O ⊂ K such that for all x ∈ K×,we have x ∈ O or x−1 ∈ O.

RemarkIf O is a valuation ring on K and O′ ⊂ K is a subring with O ⊂ O′, thenO′ is a valuation ring on K .

ExampleIf (K , v) is a valued field, then the set Ov := {x ∈ K | v(x) ≥ 0} is avaluation ring on K .

(In particular, O = K is the trivial valuation ring on K .)

Florian Severin (HHU Düsseldorf) Variants of Hensel’s lemma GeSAMT III, 14 July 2017 2 / ∞

Valuations and valuation rings

FactThese are the only examples: The above is a 1-to-1 correspondence ofvaluations on K (up to equivalence) and valuation rings on K.

RemarkA valuation ring Ov on K is a local ring with unique maximal ideal

mv = O \ O× = {x ∈ K | v(x) > 0}.

NotationWe denote the residue field by Kv := Ov/mv


Henselian valuations

DefinitionLet (K , v) be a valued field and let L/K be a field extension. A valuationw on L with w � K = v (or equivalently: Ow ∩ K = Ov ) is called anextension of v to L.

Theorem (Chevalley)“Extensions always exist.”

DefinitionA valuation on a field K is henselian, if it can be uniquely extended to thealgebraic closure K alg of K .

A field is henselian, if it admits a non-trivial henselian valuation.


Characterizations of Henselianity

TheoremLet (K , v) be a valued field. Then the following are equivalent:

1 v is henselian.2 v uniquely extends to the separable closure K sep of K.3 v uniquely extends to all Galois extensions of K.4 Let f ∈ Ov [X ] such that resKv (f ) ∈ Kv [X ] has a simple root α ∈ Kv.

Then there is a unique root a ∈ Ov of f with a + mv = α.5 Every polynomial f ∈ Ov [X ] of the form

f = X d + X d−1 + ad−2X d−2 + · · ·+ a0

with d ≥ 1 and a0, . . . , ad−2 ∈ mv has a root in KOv .


Variants of Henselianity

General IdeaRestrict the statements in the above characterizations of Henselianity tocertain classes of field extensions / polynomials. weaker notion(s) of Henselianity

Examples1 henselian: K sep resp. all GE / polynomials f (+ conditions)2 2-henselian: GE of even degree / ?3 p-henselian: K (p) resp. GE of degree pk / f splitting in K (p)4 Ω-henselian: (normal algebraic) Ω ⊃ K / ?


Another example: n≤-henselian valuations

DefinitionA valuation v is n≤-henselian if every f ∈ Ov [X ] of the form

f = X d + X d−1 + ad−2X d−2 + · · ·+ a0,

with d ≥ 1 and a0, . . . , ad−2 ∈ mv of degree d ≤ n has a root in K .

A field is n≤-henselian, if it admits a non-trivial n≤-henselian valuation.

RemarkEquivalently, we could consider polynomials of the form

f = X d + ad−1X d−1 + ad−2X d−2 + · · ·+ a0

with 1 ≤ d ≤ n, a0, . . . , ad−2 ∈ mv and ad−1 6∈ mv .


“Characterizations” of n≤-Henselianity

TheoremLet (K , v) be a valued field. Then for the conditions

1 v uniquely extends to all Galois extensions of polynomial-degree atmost n.

2 Let f ∈ Ov [X ] with deg f ≤ n such that resKv (f ) ∈ Kv [X ] has asimple root α ∈ Kv. Then there is a unique root a ∈ Ov of f witha + mv = α.

3 v is n≤-henselian.4 v extends uniquely to all Galois extensions of degree at most n.

we have (1) ⇒ (2) ⇒ (3) ⇒ (4).


The polynomial-degree of an extension

DefinitionThe polynomial-degree of a finite Galois extension L/K is the naturalnumber

[L : K ]poly := min {deg f | f ∈ K [X ] is irreducible andL is the splitting field of f }.

We write K≤(d) for the composite field of all Galois extensions L of Kwith polynomial-degree at most d .

RemarkWe have [L : K ]poly ≤ [L : K ] ≤ [L : K ]poly! for all Galois extensions L/K .


“Characterizations” of n≤-Henselianity, continued

CorollaryFor the following conditions, we have (4’) ⇒ (1).

4’ v extends uniquely to all Galois extensions of degree at most n!.1 v uniquely extends to all Galois extensions of polynomial-degree at

most n.

Theorem (essentially Fehm-Jahnke ’14)For each n ≥ 2, there is a valued field (K , v) which is n≤-henselian, butnot n!≤-henselian(2n)≤-henselian.

(In fact, v can be chosen not to be p≤-henselian for any prime p > n.)


The topology induced by a valuation

DefinitionLet (K , v) be a valued field. The topology induced by v is the topology onK with base

{a · Ov + b | a, b ∈ K , a 6= 0}.

Equivalently, it is the topology with base

{a ·mv + b | a, b ∈ K , a 6= 0}.

Note that a · Ov + b = {x ∈ K | v(x − b) ≥ v(a)}.

Random factTwo basic open sets intersect iff one is contained in the other.


The unique n≤-henselian topology

TheoremLet K 6= K sep and n� 0 (where the bound depends on K). Then everytwo non-trivial n≤-henselian valuations on K induce the same topology.

The proof uses a similar result about p-henselian valuations, due to JochenKoenigsmann (’95).

Note that the Theorem is a stronger version of the same result forhenselian valuations:

Corollary (Endler/Engler ’75/’76; F.K. Schmidt ’33)Let K 6= K sep. Then every two non-trivial henselian valuations on Kinduce the same topology.


The canonical n≤-henselian valuation

Fix some field K and let H≤n(K ) denote the set of n≤-henselian valuationrings on K .

NotationFor n ∈ N, let dn := max {d ∈ N | d! ≤

√n}.

DefinitionWe partition the set of n≤-henselian valuations in two classes:

H≤n1 (K ) = {Ov ∈ H≤n(K ) | Kv≤(dn) 6= Kv}

H≤n2 (K ) = {Ov ∈ H≤n(K ) | Kv≤(dn) = Kv}


Structure of H≤n1 (K ) and H≤n2 (K )

FactH≤n1 (K ) is linearly ordered by inclusion, and for O1 ∈ H

≤n1 (K ) and

O2 ∈ H≤n2 (K ) we have O1 ⊃ O2.

FactO? :=

⋂H≤n1 (K ) is an n≤-henselian valuation ring.

FactIf H≤n2 (K ) 6= ∅, it has a unique maximal element O? (w.r.t. ⊂).



DefinitionIf H≤n2 (K ) 6= ∅, let O≤n := O?. Otherwise, let O≤n := O?.The valuation O≤n is called the canonical n≤-henselian valuation on K .

RemarkFor O1 ∈ H≤n1 (K ) and O2 ∈ H

≤n2 (K ), we have O1 ⊃ O≤n ⊃ O2. In

particular, the set {O≤n | n ∈ N} is linearly ordered by inclusion.

FactLet v denote the canonical henselian valuation on K. Then Ov =

⋂O≤n if

Kv is separably closed, and OK =⋃

n�0O≤n if it is not.



Let m(K ) := min {[L : K ] | L/K is a proper Galois extension}.

PropositionSuppose that K is n≤-henselian. Then (1) ⇒ (2) ⇒ (3) for

1 n ≥ (m(K )!)22 The canonical n≤-henselian valuation is non-trivial.3 n ≥ m(K )2.

The proof needs the following

LemmaIf Kv has a proper GE of polynomial-degree at most n, so does K.



Proof (of the Proposition).Note that O≤n is trivial iff H≤n1 (K ) = ∅ (or H≤n(K ) = {K}).

For (1) ⇒ (2), let (m(K )!)2 ≤ n and recall dn = max {d | (d!)2 ≤ n}.Then dn ≥ m(K ), so K has a proper Galois extension of degree at mostdn. Hence K≤(dn) 6= K , and thus K ∈ H≤n1 (K ).

For (2) ⇒ (3), let n < m(K )2 and note that dn! ≤√

n < m(K ), so K hasno proper Galois extension of degree ≤ dn!, and hence it has no properGalois extension of polynomial-degree ≤ dn. By the Lemma, this impliesKv≤(dn) = Kv for all valuations v on K , hence H≤n1 (K ) = ∅.


An application to model theory

PropositionLet K 6= K sep and v a non-trivial henselian valuation on K with Ovdefinable in Lring = {0, 1,+, ·}. Then there is a ∅-definable non-trivialvaluation ring O ⊂ K inducing the unique henselian topology on K.

Remark (Jahnke-Koenigsmann ’14)If we replace “induces the henselian topology” by “is henselian”, thestatement is false.



Proof.By work of Jahnke and Koenigsmann, we can assume that Kv is notseparably closed (otherwise, there is an ∅-definable henselian valuation), sothere is a Galois extension L/Kv of degree n ≥ 2.

For m := (n!)2, we have [L : Kv ]poly ≤ [L : Kv ] = n = dm, soOv ∈ H≤m1 (K ). Thus, by definition of the canonical m≤-henselianvaluation, we have O≤m ⊂ Ov .

Now let φ(x , b) define Ov for some b ∈ K `, i.e.,

φ(K , b) = {x ∈ K | K |= φ(x , b)} = Ov .

Consider the set S := {c ∈ K ` | φ(K , c) ∈ H≤m1 (K )} of parameters c forwhich φ(x , c) defines an m≤-henselian valuation w on K withKw≤(dm) 6= Kw . The set S is ∅-definable!



Proof (continued) ...Recall that S = {c ∈ K ` | φ(K , c) ∈ H≤m1 (K )} and observe that

O :=⋂c∈S

φ(K , c) = {x ∈ K | ∀c ∈ S φ(x , c)}

is an ∅-definable valuation ring on K (since S is ∅-definable andO≤m ⊂ O).

As φ(K , b) = Ov ∈ H≤m1 (K ), we have b ∈ S, and hence O ⊂ Ov . ThusOOv = Ov 6= K , which implies the claim.


Variants of Henselianity, compared

henselian

� �� P-henselian

?+3 N≤-henselian

ks +3t-henselian

?ks

WhereP-henselian is short for “p-henselian for all primes p”,N≤-henselian is short for “n≤-henselian for all n ∈ N”, andt-henselian means “being elementarily equivalent to some henselianfield”.

In general, none of these imply Henselianity. (But if K is neither separablyclosed, nor real closed and GK is pro-soluble, the latter do.)


ValuationsHenselian valuations n/henselian valuationsAn application to model theory

variants of hensel's lemma - hhureh.math.uni-duesseldorf.de/~severin/docs/hensel-gesamt... · 2017....

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