Erdős problem 90

Sawin's Lemmas 11–12 / Remarks Proposition 2.3: the totally real tower.

There exist $rdBound : \mathbb{R}$ and a *single* infinite set $Q$ of rational primes $q \equiv 1 \pmod 4$ such that for every $N$ one can find a totally real number field $F/\mathbb{Q}$ of degree $\ge N$ with bounded root discriminant $|disc F|^{1/[F:\mathbb{Q}]} \le rdBound$ in which *every* prime $q \in Q$ splits completely.

The load-bearing feature is the quantifier order: $Q$ is fixed *before* $F$, so the same primes split completely in fields of *unbounded* degree. (For a single fixed $F$, Chebotarev already gives infinitely many completely split primes $\equiv 1 \pmod 4$, so a per-field statement would be vacuous.) This uniform splitting in an unbounded tower is the key arithmetic input to the disproof. It is proved as Lemmas 11–12 of Sawin, [arXiv:2605.20579](https://arxiv.org/abs/2605.20579), and as Proposition 2.3 of the [Remarks](https://arxiv.org/abs/2605.20695) paper, via the Golod–Shafarevich inequality for pro-$2$ groups together with the Hajir–Maire–Ramakrishna (2003) tower construction.

A "completely split" rational prime $q$ in $F$ is one for which $(q)$ is the product of exactly $[F:\mathbb{Q}]$ distinct maximal ideals.