Sawin’s totally real towers
The main theorem constructs totally real number fields of arbitrarily large degree with root discriminant at most 255255. One fixed infinite set of rational primes, all congruent to one modulo four, splits completely in every field. The degree can therefore grow while both the discriminant bound and the splitting conditions remain fixed.
The construction starts with the maximal totally real pro-2 extension of the rationals unramified outside six odd primes. Carefully chosen Frobenius relations force primes to split, while a Golod–Shafarevich estimate keeps the resulting quotient infinite. Its finite layers provide fields of large degree, and restricted ramification controls their discriminants. These results formalize the arithmetic tower used in Sawin’s unit-distance work.
Selected theorems and definitions
- definition
The maximal real pro-p extension with restricted ramification
The ambient arithmetic extension is built from its admissible finite Galois layers.
- theorem
Which finite fields lie in the maximal extension?
Admissibility is equivalent to containment in the constructed maximal extension.
- theorem
Five generators for the initial arithmetic group
The maximal real pro-2 group for the six chosen ramified primes has generator rank exactly five.
- theorem
A five-generator presentation with at most six relations
The initial arithmetic group admits an actual finite minimal pro-2 presentation.
- theorem
Primes with deep Frobenius lifts
Arithmetic Frobenius elements can be lifted to successively deeper levels of one free pro-2 presentation.
- theorem
Killing infinitely many Frobenius elements without making the group finite
There is an infinite quotient in which the selected primes have trivial decomposition groups.
- theorem
A uniform root-discriminant bound of 255255
Restricted tame ramification gives an exponential bound for the actual number-field discriminant.
- theorem
One infinite prime set splitting in fields of unbounded degree
A single infinite set of primes splits completely in totally real fields with uniformly bounded root discriminant.
- theorem
A Martinet-type bounded-discriminant corollary
Forgetting the splitting conditions yields totally real number fields of unbounded degree and bounded root discriminant.
- theorem
Shafarevich’s relation-rank bound for odd primes
Degree-two continuous cohomology of the maximal everywhere-unramified pro-p group is finite and arithmetically bounded.
References used in these guides
- Will Sawin. An explicit lower bound for the unit distance problem. 2026. The arithmetic tower ingredients are related to Lemmas 11–12; the library does not formalize the whole unit-distance argument.
- Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, and Melanie Matchett Wood. Remarks on the disproof of the unit distance conjecture. 2026. Proposition 2.3 states the bounded-root-discriminant totally real tower with infinitely many completely split primes congruent to one modulo four.
- Farshid Hajir, Christian Maire, and Ravi Ramakrishna. Cutting towers of number fields. 2019. Cited as the arXiv preprint; develops cutting methods with refined Golod–Shafarevich estimates.
- E. S. Golod and I. R. Shafarevich. On the class field tower. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 28 (2), 261–272, 1964. Original article in Russian.
- Igor R. Šafarevič. Extensions à points de ramification donnés (en russe). Publications Mathématiques de l’IHÉS, Institut des Hautes Études Scientifiques, 18, 71–92, 1963.
- Jacques Martinet. Tours de corps de classes et estimations de discriminants. Inventiones mathematicae, 44, 65–73, 1978.
- Luis Ribes and Pavel Zalesskii. Profinite Groups. Springer-Verlag Berlin Heidelberg, 2010. Second edition; background for Frattini quotients, generator rank, and pro-p presentations.