I would like to write about a new paper ([AJ15]) which Franziska Jahnke and I have written and put on the arXiv. It’s called Henselianity in the language of rings. In it we investigate the relationship between the following four properties of a field :
(h) is henselian, i.e. admits a nontrivial henselian valuation,
(eh) is elementarily henselian, i.e. every is henselian,
(def) admits a definable nontrivial henselian valuation, and
(-def) admits a -definable nontrivial henselian valuation.
By `definable’ we mean `definable in the language of rings’. Our paper is the latest work on this topic, which began with the work of Prestel and Ziegler who show, in [PZ78], that there exist t-henselian, nonhenselian fields. Subsequently, several more recent papers explore this topic and related issues of definable valuations; examples include [JK15] and [FJ15]. Our work builds on these three papers, and others. A more complete survey of the literature is available in our paper.
There are some trivial implications between the above properties:
Franziska and I investigate whether any other implications hold between these properties across the class of all fields of characteristic zero, and across the smaller classes
, for a prime;
where denotes the canonical henselian valuation on . Our main theorem gives the `complete picture’ for each of these classes
Main Theorem (Theorem 1.1, [AJ15])
- In the class the complete picture is
- For each prime , in the class the complete picture is
- Consequently, in the class the complete picture is simply given by (1) and (2), above.
Given the `trivial’ implications, (a) and (b) above, the proof of Theorem 1.1 boils down to showing the following:
- In the class the implication holds.
- For each prime , in the class the implications and hold.
- No implications hold other than the ones listed above and the `trivial’ ones.
The proof of (4) has two stages: first we show that using Theorem B from [JK15]; then we employ the Omitting Types Theorem to remove the parameter.
The proof of (5) is more tricky. The implication follows from basic properties of the canonical henselian valuation combined with some model-theoretic trickery. Franziska and I call these arguments `Sirince tricks’ because we wrote them down when we were on a research visit to the wonderful Nesin Mathematics Village, near Sirince, Turkey! The proof of the implication depends on some delicate arguments involving uniform definitions of canonical -henselian valuations.
Finally, the proof of (6) is a set of four counterexamples. The first (given by Prestel-Ziegler, [PZ78]) simply shows the existence (in equicharacteristic zero) of t-henselian nonhenselian fields. The second and third examples were given in [JK15] and are based on the construction in [PZ78]. Finally, the fourth example is based on a more sophisticated construction give in [FJ15] of t-henselian nonhenselian fields with various properties – we extend it to defectless fields of positive characteristic.
There are several key open questions. In my opinion, perhaps the most prominent is: can our results be extended to equicharacteristic ?
[AJ15] Sylvy Anscombe and Franziska Jahnke. Henselianity in the language of rings. Manuscript, 2015. arXiv
[FJ15] Arno Fehm and Franziska Jahnke. On the quantifier complexity of definable canonical henselian valuations. Mathematical Logic Quarterly, 61(4-5):347-361, 2015.
[JK15] Franziska Jahnke and Jochen Koenigsmann. Defining coarsenings of valuations. To appear in the Proceedings of the Edinburgh Mathematical Society, 2015. arXiv
[PZ78] Alexander Prestel and Martin Ziegler. Model-theoretic methods in the theory of topological fields. Journal für die reine und angewandte Mathematik, 299(300):318-341, 1978.