Question

Show that Q(\root(3)(2)) has only the identity automorphism. Show that $\mathbb{Q}(\sqrt[3]{2})$ has only the identity automorphism.

Name: show that qroot32 has only the identity automorphism show that mathbbqsqrt32 has only the identity automorphism 95128
Uploaded: 2023-02-26T13:40:03-08:00
Duration: 2 min 10 s
Channel: Sri K
Description: show that qroot32 has only the identity automorphism show that mathbbqsqrt32 has only the identity automorphism 95128

          Show that Q(\root(3)(2)) has only the identity automorphism.
Show that $\mathbb{Q}(\sqrt[3]{2})$ has only the identity automorphism.

Added by Susan D.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Sri K

02/26/2023

Step 1

The minimal polynomial of $\sqrt[3]{2}$ over $\mathbb{Q}$ is $x^3 - 2$. The roots of $x^3 - 2$ are $\sqrt[3]{2}, \omega \sqrt[3]{2}, \omega^2 \sqrt[3]{2}$, where $\omega = e^{2\pi i/3} = \frac{-1 + i\sqrt{3}}{2}$ is a primitive cube root of unity. Since $\omega Show more…

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