10. Let F(a) be the field described in Exercise 8. Show that a² and a² + a are zeros of x³ + x + 1.
10. Let F(a) be the field described in Exercise 8. Show that a² and a² + a are zeros of x³ + x + 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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I'm looking for the answer to #10. Please include rationale for the solution. Thanks!
![**Exercise 8:**
Let \( F = \mathbb{Z}_2 \) and let \( f(x) = x^3 + x + 1 \in F[x] \). Suppose that \( a \) is a zero of \( f(x) \) in some extension of \( F \). How many elements does \( F(a) \) have? Express each member of \( F(a) \) in terms of \( a \). Write out a complete multiplication table for \( F(a) \).
**Exercise 10:**
Let \( F(a) \) be the field described in Exercise 8. Show that \( a^2 \) and \( a^2 + a \) are zeros of \( x^3 + x + 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F174d7d89-86f6-4fd3-817b-704de3f40cca%2Fab3eaec8-de6b-4349-83b0-ae7b9b96e43c%2Fh430r5j_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 8:**
Let \( F = \mathbb{Z}_2 \) and let \( f(x) = x^3 + x + 1 \in F[x] \). Suppose that \( a \) is a zero of \( f(x) \) in some extension of \( F \). How many elements does \( F(a) \) have? Express each member of \( F(a) \) in terms of \( a \). Write out a complete multiplication table for \( F(a) \).
**Exercise 10:**
Let \( F(a) \) be the field described in Exercise 8. Show that \( a^2 \) and \( a^2 + a \) are zeros of \( x^3 + x + 1 \).
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