Henselian Elements Franz-viktor Kuhlmann and Josnei Novacoski with an Appendix by Hagen Knaf

Henselian Function Fields

Abhyankar places admit local uniformization in any characteristic

We prove that every place P of an algebraic function field F |K of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field FP over K is equal to the transcendence degree of F |K, and the extension FP |K is separable. We generalize this result to the case where P dominates a regular loca...

Dense subfields of henselian fields, and integer parts

We show that every henselian valued field L of residue characteristic 0 admits a proper subfield K which is dense in L. We present conditions under which this can be taken such that L|K is transcendental and K is henselian. These results are of interest for the investigation of integer parts of ordered fields. We present examples of real closed fields which are larger than the quotient fields o...

Elementary Properties of Power Series Fields over Finite Fields

In spite of the analogies between Qp and Fp((t)) which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for Qp to the case of Fp((t)) does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on Fp((t)). We formulate an elementary property...

Additive Polynomials and Their Role in the Model Theory of Valued Fields

We discuss the role of additive polynomials and p-polynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of valued fields of positive characteristic as modules over such rings. We prove the existence of Frobenius-closed bases of algebraic function fields F |K in one ...