February 9
Russell Miller, CUNY
Properties of Generic Algebraic Fields

The algebraic field extensions of the rational numbers $\mathbb{Q}$ – equivalently, the subfields of the algebraic closure $\overline{\mathbb{Q}}$ – naturally form a topological space homeomorphic to Cantor space. Consequently, one can speak of 'large' collections of such fields, in the sense of Baire category: collections that are comeager in the space. Under a standard definition, the 1-generic fields form a comeager set in this space. Therefore, one may think of a property common to all 1-generic fields as a property that one might reasonably expect to be true of an arbitrarily chosen algebraic field.

We will present joint work with Eisenträger, Springer, and Westrick that proves several intriguing properties to be true of all 1-generic fields $F$. First, in every such $F$, both the subring $\mathbb{Z}$ of the integers and the subring $\mathcal{O}_F$ of the algebraic integers of $F$ cannot be defined within $F$ by an existential formula, nor by a universal formula. (Subsequent work by Dittman and Fehm has shown that in fact these subrings are completely undefinable in these fields.) Next, for every presentation of every such $F$, the root set

$ R_F = \{ p\in \mathbb{Z}[X]:p(X)=0\text{ has a solution in }F\} $

is always of low Turing degree relative to that presentation, but is essentially always undecidable relative to the presentation. Moreover, the set known as Hilbert's Tenth Problem for $F$,

$ HTP(F) = \{ p\in \mathbb{Z}[X_1,X_2,\ldots]:p(X_1,\ldots,X_n)=0\text{ has a solution in }F^n\}, $

is exactly as difficult as $R_F$, which is its restriction to single-variable polynomials. Finally, even the question of having infinitely many solutions,

$\{ p\in \mathbb{Z}[X_1,X_2,\ldots]:p(\vec{X})=0\text{ has infinitely many solutions in }F^n\}, $

is only as difficult as $R_F$. These results are proven by using a forcing notion on the fields and showing that it is decidable whether or not a given condition forces a given polynomial to have a root, or to have infinitely many roots.