Computability and the absolute Galois group of the rational numbers

The absolute Galois group Gal(Q) is well-known throughout mathematics. Its elements are precisely the symmetries of the algebraic closure of the rational numbers. In practice, though, this group is particularly difficult to study.

There are continuum-many of these symmetries, most of which cannot be computed by any computer (or Turing machine) running any finite-length program whatsoever. However, the symmetries that mathematicians encounter on a regular basis are essentially always computable -- perhaps because these are fundamental to the group, or perhaps just because noncomputable symmetries are naturally more difficult to examine and work with.

This project aims to determine just how much difference there is between the computable symmetries (as a group) and the larger group of all symmetries. The research work lies at the interface of logic and number theory and is likely to attract the interest of both communities. Graduate students from CUNY Graduate Center will participate in this project.

An analogous situation exists with the field of all real numbers: only countably many real numbers have computable decimal expansions, so the vast majority of real numbers are noncomputable, yet the computable ones are the only ones ever encountered in daily life. Here, it is known that the computable real numbers form a subfield extremely similar to the full field of all real numbers, an elementary subfield with exactly the same first-order properties.

This grant will fund research to attempt to determine whether Gal(Q) is analogous in this way: do the computable symmetries form an elementary subgroup of the full group? (Or, at a minimum, are the two elementarily equivalent?) If so, then mathematicians should be able to determine many results about the full group just by examining the computable symmetries, which are far more accessible. If not, that would suggest that the absolute Galois group is a thornier object than the field of real numbers, with its noncomputable symmetries somehow essential to its character.

However, even then, it is possible that the subgroup might be elementary for relatively simple properties (e.g., purely existential statements about the group), in which case this project will attempt to find the first level at which the subgroup stops imitating the full group.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.