Explanation Given Information: Euclidean Algorithm f ( x ) and g ( x ) are nonzero polynomials in F . g ( x ) = f ( x ) q 0 ( x ) + r 1 ( x ) deg r 1 ( x ) < deg f ( x ) f ( x ) = r 1 ( x ) q 1 ( x ) + r 2 ( x ) deg r 2 ( x ) < deg r 1 ( x ) r 1 ( x ) = r 2 ( x ) q 2 ( x ) + r 3 ( x ) deg r 3 ( x ) < deg r 2 ( x ) ⋮ ⋮ r n − 2 ( x ) = r n − 1 ( x ) q n − 1 ( x ) + r n ( x ) deg r n ( x ) < deg r n − 1 ( x ) r n − 1 ( x ) = r n ( x ) q n ( x ) r n ( x ) is the last nonzero remainder. Formula Used: ( 1 ) If f ( x ) , and g ( x ) are polynomials in F and f ( x ) | g ( x ) , then there exists a polynomial p ( x ) in F such that g ( x ) = p ( x ) f ( x ) . ( 2 ) Let f ( x ) and g ( x ) be nonzero polynomials in F [ x ] , A polynomial d ( x ) in F [ x ] is the greatest common divisor of f ( x ) and g ( x ) if these conditions are satisfied: 1. d ( x ) is a monic polynomial, 2. d ( x ) | f ( x ) and d ( x ) | g ( x ) , and 3. h ( x ) | f ( x ) and h ( x ) | g ( x ) imply that h ( x ) | d ( x ) . Proof: Consider the nonzero polynomials f ( x ) and g ( x ) in F [ x ] . g ( x ) = f ( x ) q 0 ( x ) + r 1 ( x ) deg r 1 ( x ) < deg f ( x ) f ( x ) = r 1 ( x ) q 1 ( x ) + r 2 ( x ) deg r 2 ( x ) < deg r 1 ( x ) r 1 ( x ) = r 2 ( x ) q 2 ( x ) + r 3 ( x ) deg r 3 ( x ) < deg r 2 ( x ) ⋮ ⋮ r n − 2 ( x ) = r n − 1 ( x ) q n − 1 ( x ) + r n ( x ) deg r n ( x ) < deg r n − 1 ( x ) r n − 1 ( x ) = r n ( x ) q n ( x ) r n ( x ) is the last nonzero remainder

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ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 8.2, Problem 31E

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To prove: That in the description of Euclidean Algorithm a−1rn(x) is the greatest common divisor of f(x) and g(x).

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Chapter 8.2, Problem 30E

Chapter 8.2, Problem 32E

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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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To prove: That in the description of Euclidean Algorithm a − 1 r n ( x ) is the greatest common divisor of f ( x ) and g ( x ) .

Solution Summary: The author explains that Euclidean Algorithm a-1r_n(x) is the greatest common divisor of the two nonzero polynomials

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To prove: That in the description of Euclidean Algorithm a−1rn(x) is the greatest common divisor of f(x) and g(x).

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Chapter 8 Solutions