Uniform definability of henselian valuation rings in the Macintyre language

We discuss definability of henselian valuation rings in the Macintyre language $\mathcal{L}_{\rm Mac}$, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$, and henselian valuation rings with value group $\mathbb{Z}$ are uniformly $\exists\forall$-$\emptyset$-definable in the ring language, but not uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$. We apply these results to local fields $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$, as well as to higher dimensional local fields

De Morgan's law and the theory of fields

We show that the classifying topos for the theory of fields does not satisfy De Morgan's law, and we identify its largest dense De Morgan subtopos as the classifying topos for the theory of fields of nonzero characteristic which are algebraic over their prime fields