Explicit Approximations to Class Field Towers

TL;DR

This work constructs explicit, efficiently computable infinite families of number fields with subpolynomial growth in the root discriminant, answering Peikert–Rosen’s question. The approach builds large-degree subfields arising from two-elementary class field towers attached to real quadratic fields and combines algebraic genus theory with analytic density results: conditional Chebotarev per PTW and automorphic L-function bounds per KM. A core step is proving a nontrivial lower bound on the number of ‘useful’ prime pairs $S$, which ensures the resulting Galois 2-extensions yield $D_N^{1/n} = O(n^{\varepsilon})$ for all $\varepsilon>0$, with unconditional and conditional ingredients. The paper also clarifies connections to Rédei symbols and provides an explicit example $N'$ with $[N':\mathbb{Q}]=256$ and $D_{N'}^{1/256}\approx 55.58$, demonstrating the construction in a concrete setting and its potential cryptographic relevance.

Abstract

We answer a question of Peikert and Rosen by giving for each $ε> 0$ an efficient construction of infinite families of number fields $N$ such that the root discriminant $D_N^{1/[N:\mathbb{Q}]}$ is bounded above by a constant times $[N:\mathbb{Q}]^ε$.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Lemma 2.4
• proof