Exercise 7.1.17 (Geometric description of Galois group)

Question

Draw a diagram showing the three roots of $t^3-2$ and the elements of $H = \mathrm{Gal}(\mathbb{Q}(\xi,\omega) : \mathbb{Q}(\omega))$ acting on them. There is a simple geometric description of the elements of $\mathrm{Gal}(\mathbb{Q}(\xi,\omega) : \mathbb{Q})$ that belong to the subgroup $H$. What is it?

Richard Ganaye · Accepted Answer

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

\begin{proof} 
The points $A,B,C$ of the Argand-Cauchy plane with affixes $\xi, \omega \xi, \omega^2 \xi$ form an equilateral triangle $T = \{A,B,C\}$.

$H = \Gal(\Q(\xi,\omega) : \Q(\omega))$ is generated by $\sigma$, such that $\sigma(\xi) = \omega \xi, \sigma(\omega \xi) = \omega^2 \xi, \sigma(\omega^2 \xi) = \xi$. The corresponding transformation of the plane is $ho : M(z) \mapsto M'(\omega z)$, wich is the rotation with center $O(0)$ and angle $2\pi/3$.

So $H \simeq H' = \{\mathrm{id}, ho, ho^2\}$, group of the rotations which send the equilateral triangle $T$ on itself.

Note: similarly, the geometric description of the elements of the group $G = \Gal(\Q(\omega, \xi) : \Q)$ is the full isometry group of $T$ ($3$ rotations, and $3$ reflection symmetries).

\end{proof}

Exercise 7.1.17 (Geometric description of Galois group)

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Exercise 7.1.17 (Geometric description of Galois group)

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