AKE-principles for deeply ramified fields
Jonas van der Schaaf

Abstract:
In this thesis we will expand upon a result of Jahnke and Kartas. They obtain Ax-Kochen/Ershov (AKE) principles for perfectoid fields and their tilts. Their methods all take place in a setting with only perfect fields (in particular perfectoid fields). We extend some of their results to the deeply ramified (non-perfect) setting.

In chapter 2, we will give an introduction to (valued) field theory, as well as introduce some more advanced theorems which we will use in later parts of the thesis. Next, in chapter 3, we exhibit a proof of the AKE principle for separably tame valued fields for fields of possibly infinite p-degree.

Then in chapter 4, we will introduce our own work: we construct an elementary class which will serve as the class for which we exhibit our AKE principle. In particular, we prove the proposition "Let (K, v, t) be a pointed valued field of equal characteristic p>0 which is henselian and deeply ramified. If t ∈ m_v\{0} then the property of O_v[1/t] being separably algebraically maximal is elementary."

The class of pointed valued fields with this property is the class we are interested in. We will justify this in the second half of chapter 4, and give several ways of recognizing valued fields in the class.

Finally, in chapter 5 we will prove our various AKE principles. Most theorems we prove will be variants of the following theorem:
  Let (K, v) ⊆ (L0, w0), (L1, w1 ) be extensions of henselian deeply ramified valued fields such that
   (i)  L0/K and L1/K are separable extensions
   (ii) L0 and L1 have the same (possibly infinite) p-degree
   (iii) there is some t ∈ K × such that vt > 0 and the valuation rings O_v[1/t], O_w0[1/t] O_w1[1/t] are all algebraically maximal.
 Then the following are equivalent:
   (i) (L0, w0) ≡(K,v) (L1, w1 ) in the language of valued fields,
   (ii) w0L0 ≡vK w1L1 and O_w0/t ≡O_v/t O_w1/t in the language of ordered abelian groups and rings respectively.

In particular, we will show this result for elementary equivalence, elementary embedding, a modified version without base field, and finally the same result for existential embeddings.