Exercise 9.1.10 (Eigenvectors of automorphisms)

Question

What does the proof of Lemma 9.1.8 tell you about the eigenvectors and eigenvalues of the elements of $\mathrm{Gal}_K(t^n -a)$ ?

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}}

Let $\varphi \in \Gal_K(t^n -a)$.

By the proof of Lemma 9.1.8, we know that $\varphi(\xi)/\xi \in K$, where $\xi$ is any root of $t^n -a$ in $M = \mathrm{SF}_K(t^n -a)$.

Therefore $\varphi(\xi) = \lambda \xi$ for some $\lambda \in K$, and $\varphi : M 	o M$ is $K$-linear, so that all the roots of $t^n -a$ are eigenvectors of $\varphi$, and the corresponding eigenvalues are in $K$.

Exercise 9.1.10 (Eigenvectors of automorphisms)

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Exercise 9.1.10 (Eigenvectors of automorphisms)

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