Seminar

The subfields of the algebraic closure $\overline{\mathbb Q}$ of the rational numbers are known to form a topological space in a natural way.  Since this space is homeomorphic to Cantor space, one may apply the notions of Baire category and of genericity to it.  The generic subfields form a comeager subset of the space.

In this context, there is a natural notion of forcing.  We show that it is decidable whether a given forcing condition in this notion forces a given existential formula, and also whether it forces the negation of that formula.  This allows us to prove results holding of all generic subfields $F$ of $\overline{\mathbb Q}$:  for such fields, Hilbert's Tenth Problem $HTP(F)$ is in general not decidable from the atomic diagram of a presentation of $F$, but it is only as difficult as its restriction to polynomial equations in a single variable over $F$,  Moreover, there do exist sets that are computably enumerable relative to $F$, but not diophantine in $F$.  We also infer results about the undefinability of coinfinite non-thin subsets of $\mathbb Q$ by universal formulas in such fields $F$. Since the generic fields form a comeager class, all these properties may be considered to hold in ``almost all'' subfields of $\overline{\mathbb Q}$.

This is joint work with Kirsten Eisentr\"ager, Caleb Springer, and Linda Westrick.

Forcing in Algebraic Field Extensions of the Rationals 166 KB application/pdf

Forcing In Algebraic Field Extensions Of The Rationals