Question

7. Let $\sigma$ be the automorphism of $\mathbb{Q}(\pi)$ that maps $\pi$ onto $-\pi$. a. Describe the fixed field of $\sigma$. b. Describe all extensions of $\sigma$ to an isomorphism mapping the field $\mathbb{Q}(\sqrt{\pi})$ onto a subfield of $\overline{\mathbb{Q}(\pi)}$.

Name: 7 let sigma be the automorphism of mathbbqpi that maps pi onto pi a describe the fixed field of sigma b describe all extensions of sigma to an isomorphism mapping the field mathbbqsqrtpi ont 39394
Uploaded: 2024-01-16T03:29:35-08:00
Duration: 2 min 2 s
Channel: Nick Johnson
Description: 7 let sigma be the automorphism of mathbbqpi that maps pi onto pi a describe the fixed field of sigma b describe all extensions of sigma to an isomorphism mapping the field mathbbqsqrtpi ont 39394

          7. Let $\sigma$ be the automorphism of $\mathbb{Q}(\pi)$ that maps $\pi$ onto $-\pi$.
a. Describe the fixed field of $\sigma$.
b. Describe all extensions of $\sigma$ to an isomorphism mapping the field $\mathbb{Q}(\sqrt{\pi})$ onto a subfield of $\overline{\mathbb{Q}(\pi)}$.

7 let sigma be the automorphism of mathbbqpi that maps pi onto pi a describe the fixed field of sigma b describe all extensions of sigma to an isomorphism mapping the field mathbbqsqrtpi ont 39394

Added by Kenneth T.

Elementary and Intermediate Algebra

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

Instant Answer

Solved by Expert Nick Johnson

01/16/2024

Step 1

Let $F = \mathbb{Q}(\pi)$. The automorphism $\sigma: F \to F$ is defined by $\sigma(\pi) = -\pi$. Since $\sigma$ is an automorphism of $\mathbb{Q}(\pi)$, it must fix the rational numbers $\mathbb{Q}$. Let $f(\pi) \in \mathbb{Q}(\pi)$ be an element in the fixed Show more…

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7. Let $\sigma$ be the automorphism of $\mathbb{Q}(\pi)$ that maps $\pi$ onto $-\pi$. a. Describe the fixed field of $\sigma$. b. Describe all extensions of $\sigma$ to an isomorphism mapping the field $\mathbb{Q}(\sqrt{\pi})$ onto a subfield of $\overline{\mathbb{Q}(\pi)}$.