Explanation Formula Used: If f ( x ) is a polynomial of degree 2 or 3 over the field F , then f ( x ) is irreducible over F if and only if f ( x ) has no zeros in F . Explanation: Consider the given statement, Let F and F ′ be fields with F ⊆ F ′ . If f ( x ) is irreducible over F .then f ( x ) is irreducible over F ′ . Let F = Q , F ′ = R and a polynomial f ( x ) = x 2 − 2 of degree 2. The zeros of f ( x ) = x 2 − 2 are 2 and − 2

8th Edition

ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 8.3, Problem 9TFE

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Whether the statement, ‘Let F and F′ be fields with F⊆F′. If f(x) is irreducible over F.then f(x) is irreducible over F′.’ is true or false.

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Chapter 8.3, Problem 8TFE

Chapter 8.3, Problem 1E

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Chapter 8 Solutions

Elements Of Modern Algebra

Ch. 8.1 - True or False Label each of the following...Ch. 8.1 - Prob. 2TFE Ch. 8.1 - Prob. 3TFE Ch. 8.1 - Prob. 4TFE Ch. 8.1 - Prob. 5TFE Ch. 8.1 - Prob. 6TFE Ch. 8.1 - Prob. 7TFE Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E

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Whether the statement, ‘Let F and F ′ be fields with F ⊆ F ′ . If f ( x ) is irreducible over F .then f ( x ) is irreducible over F ′ .’ is true or false.

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Whether the statement, ‘Let F and F′ be fields with F⊆F′. If f(x) is irreducible over F.then f(x) is irreducible over F′.’ is true or false.

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Chapter 8 Solutions