a) Explanation Given Information: The set of all polynomials in F [ x ] such that the sum of the coefficients is zero. I [ x ] = { a 0 + a 1 x 1 + ⋅ ⋅ ⋅ + a n x n | a i ∈ F and a 0 + a 1 + ⋅ ⋅ ⋅ + a n = 0 } Formula Used: A subset I of a ring R is an ideal of R provided the following conditions are satisfied: 1. I is nonempty. 2. x ∈ I and y ∈ I imply that x + y ∈ I . 3. x ∈ I implies that − x ∈ I . 4. x ∈ I and r ∈ R imply that x r and r x are in I . Proof: Consider the given set I [ x ] of all polynomials in F [ x ] such that the sum of the coefficients is zero. I [ x ] = { a 0 + a 1 x 1 + ⋅ ⋅ ⋅ + a n x n | a i ∈ F and a 0 + a 1 + ⋅ ⋅ ⋅ + a n = 0 } 1. I [ x ] is non empty, as I [ x ] contains a zero polynomial. 2. Let f ( x ) = ∑ i = 0 n a i x i ∈ I [ x ] , and g ( x ) = ∑ i = 0 m b i x i ∈ I [ x ] . Since the polynomials f ( x ) and g ( x ) are taken from I [ x ] so the sum of coefficients is zero for each polynomial. ∑ i = 1 n a i = 0 and ∑ i = 1 m b i = 0 For m = n . The sum of f ( x ) and g ( x ) is given by, f ( x ) + g ( x ) = ∑ i = 1 n a i x i + ∑ i = 1 n b i x i = ∑ i = 1 n ( a i + b i ) x i ∑ i = 1 n ( a i + b i ) = ( a 0 + b 0 ) + ( a 1 + b 1 ) + … + ( a n + b n ) = ( a 0 + a 1 + … + a n ) + ( b 0 + b 1 + … + b n ) = 0 + 0 = 0 For m > n . Here a i = 0 for all i > n . The sum of f ( x ) and g ( x ) is given by, f ( x ) + g ( x ) = ∑ i = 1 m a i x i + ∑ i = 1 m b i x i = ∑ i = 1 m ( a i + b i ) x i ∑ i = 1 m ( a i + b i ) = ( a 0 + b 0 ) + ( a 1 + b 1 ) + … + ( a m + b m ) = ( a 0 + a 1 + … + a m ) + ( b 0 + b 1 + … + b m ) = 0 + 0 = 0 For n > m . Here b i = 0 for all i > m b) To determine The proof of the statement “ I [ x ] is a principal ideal” if it is true, otherwise disprove it.

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ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 8.1, Problem 9E

To determine

To prove: The set of all polynomials in F[x] such that the sum of the coefficients is zero, is an ideal of F[x].

To determine

The proof of the statement “I[x] is a principal ideal” if it is true, otherwise disprove it.

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Chapter 8.1, Problem 8E

Chapter 8.1, Problem 10E

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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To prove: The set of all polynomials in F [ x ] such that the sum of the coefficients is zero, is an ideal of F [ x ] .

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To prove: The set of all polynomials in F[x] such that the sum of the coefficients is zero, is an ideal of F[x].

The proof of the statement “I[x] is a principal ideal” if it is true, otherwise disprove it.

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Chapter 8 Solutions