An Errata for: Torsion subgroups of rational elliptic curves over the compositum of all $D_4$ extensions of the rational numbers

TL;DR

This paper revises previous results on torsion subgroups of rational elliptic curves over the compositum of all D_4 extensions of Q, correcting a misreading and clarifying the relationship between two related fields.

Contribution

It corrects a key assumption in earlier work, establishing that the fields are not necessarily equal, but the main classification results still hold.

Findings

Main results of [2] remain valid despite the correction.

The fields in question are not necessarily equal, contrary to previous claims.

The classification of torsion structures over the compositum remains accurate.

Abstract

In [2], the author claims that the fields $\mathbb{Q}(D_4^\infty)$ defined in the paper and the compositum of all $D_4$ extensions of $\mathbb{Q}$ coincide. The proof of this claim depends on a misreading of a celebrated result by Shafarevich. The purpose is to salvage the main results of [2]. That is, the classification of torsion structures of $E$ defined over $\mathbb{Q}$ when base changed to the compositum of all $D_4$ extensions of $\mathbb{Q}$ main results of [2]. All the main results in [2] are still correct except that we are no longer able to prove that these two fields are equal.