5-root of units does not belong a extension field.

eraldcoil · **Posted:** Sun, 25 Oct 2020 05:28:54 UTC

Show that there are no $a,b,c,d\in \mathbb{Q}(i)$ such that $e^{2\pi i/5}=a+b\sqrt[4]{2}+c\sqrt{2}+d(\sqrt[4]{2})^{-1}$

Previously in this problem, I found all the subfields of $\mathbb{Q}(\sqrt[4]{2},i)$ with the help of the subgroups of D_4

and Galois's Theorem as you can see in the following image:

Additionally, I have the following idea:
$e^{2\pi i /5}=a+b\sqrt[4]{2}+c(\sqrt[4]{2})^2+\frac{d}{2}(\sqrt[4]{2})^3$ with $a,b,c,d\in\mathbb{Q}(i).$ Therefore $e^{2\pi i /5}\in \mathbb{Q}(\sqrt[4]{2},i)$ .

If I could prove that $e^{2\pi i /5}\not\in \mathbb{Q}(\sqrt[4]{2},i)$ I would have the request but I don't know how to use the subfields found in the image for this (or maybe it should be solved another way)

How can proves this?

S.O.S. Mathematics CyberBoard

5-root of units does not belong a extension field.

Who is online