We study a new class of differential fields called separably differentially closed fields as a differential analogue to separably closed fields. We define them in terms of existential closedness; that is, a differential field $(K,\delta)$ of arbitrary characteristic is said to be separably differentially closed if it is existentially closed in every differential field extension that is separable (in the fieldtheoretic sense). We prove that this class of differential fields is firstorder axiomatizable in the language of differential fields (we denote this theory by $\SDCF$). We do this first by giving a full description of those prime differential ideals in $K\{x\}$ that are separable over $K$ in terms of irreducible elements of $K\{x\}$ with nonzero separant (here $(K,\delta)$ is a differential field of arbitrary characteristic and $K\{x\}$ is the differential polynomial ring over $K$ in one variable). Assuming that the extension $C_K/K^p$ is finite (here $C_K$ denotes the constants of differential field $(K,\delta)$), we then exhibit several characterizations of being separably differentially closed. In particular, one important characterization is in terms of being constrainedly closed (in the sense of Kolchin). We then observe that even after specifying the characteristic $p>0$, $\SDCF_p$ is not complete. It turns out that, in analogy to the algebraic counterpart $\SCF_p$, one only needs to specify what we call the differential degree of imperfection to describe the completions. We do this in the case of the finite degree of imperfection; namely $\SDCF_{p,\epsilon}$ for finite $\epsilon$. We also note that after adding the differential $\lambda$functions to the language (these are the suitable analogue of algebraic $\lambda$functions), one obtains a quantifier elimination result for $\SDCF^\ell_{p,\epsilon}$. Furthermore, we prove that $\SDCF^\ell_{p,\epsilon}$ is stable with unique prime model extensions. We note that our results generalize the work of Carol Wood [27], as her theory of differentially closed fields in characteristic $p>0$, denoted by $\DCF_{p}$, is a special case of ours when $\epsilon=0$; in other words, $\DCF_p=\SDCF_{p,0}$.
Date of Award  1 Aug 2023 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Omar Leon Sanchez (Supervisor) & Marcus Tressl (Supervisor) 

Model theory of separably differentially closed fields
Ino, K. (Author). 1 Aug 2023
Student thesis: Phd