The Hopf-Galois structures of the 3 and 6-division points of the lemniscate curve (Q6610542)

This paper studies the Hopf-Galois structures on the extension \(\mathbb{Q}(\varphi(\frac{\omega}{3}))/\mathbb{Q}\), where\N\[\N\omega = 2\int_0^1\frac{1}{\sqrt{1-t^4}} dt\approx 2.62206\dots\N\]\Nand \(\varphi\) is the Abel's function (Definition 1). The element \(\varphi(\frac{\omega}{3}) = \varphi(\frac{2\omega}{6})\) that is being adjoined to \(\mathbb{Q}\) is a \(6\)-division point of the lemniscate\N\[\N(x^2+y^2)^2 =x^2-y^2.\N\]\NUsing Greither-Pareigis theory, the author determines all of the Hopf-Galois structures on \(\mathbb{Q}(\varphi(\frac{\omega}{3}))/\mathbb{Q}\) -- there are exactly six of them in total (\S4.3 \& \S4.4). It was shown that the Galois correspondence for each of them is bijective (\S4.5), and as an application, a primitive element for each of the intermediate fields was also computed (\S4.6).\N\NFor the entire collection see [Zbl 1539.11005].