Explanation Formula Used: ( 1 ) If f ( x ) and g ( x ) are non zero polynomial in F [ x ] , then deg ( f ( x ) g ( x ) ) = deg f ( x ) + deg g ( x ) ( 2 ) If f ( x ) and g ( x ) are in F [ x ] and f ( x ) | g ( x ) , then there exists h ( x ) in F [ x ] such that g ( x ) = f ( x ) h ( x ) . Proof: Consider f ( x ) and g ( x ) be two nonzero polynomial in F [ x ] . Since f ( x ) | g ( x ) , therefore, for some p ( x ) in F [ x ] , g ( x ) = p ( x ) f ( x ) …… ( 1 ) Also g ( x ) | f ( x ) , therefore, for some q ( x ) in F [ x ] , f ( x ) = q

8th Edition

ISBN: 9781285965918

Author: Gilbert

Publisher: Cengage

See similar textbooks

Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

Videos

Question

Chapter 8.2, Problem 25E

To determine

To prove: If f(x) and g(x) are nonzero elements of F[x] such that f(x)|g(x) and g(x)|f(x) then f(x)=ag(x).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.2, Problem 24E

Chapter 8.2, Problem 26E

Students have asked these similar questions

6) A farmer has 60 acres on which to plant oats or corn. Each acre of oats requires 100 lbs of fertilizer and 1 hour of labor. Each acre of corn requires 50 lbs of fertilizer and 2 hours of labor. The farmer has 5000 lbs of fertilizer and 100 hours available for labor. If the profit is $60 from each acre of oats and $100 from each acre of corn, what planting combination will produce the greatest total profit? a) Fill in the following chart to help organize the information given in the problem: Oats Labor Fertilizer Land Profit b) Write down the question of interest. Corn Available c) Define variables to answer the question of interest. Call these x and y. d) Write the objective function to answer the question of interest. e) List any constraints given in the problem.

I need help with number 5.

3) Use the following system of linear inequalities graphed below to answer the questions. a) Use the graph to write the symbolic form of the system of linear inequalities. b) Is (-4,2) a solution to the system? Explain. 5 -7 -5 -3 -2 0 2 3 4 $ 6 -2 -6 -7

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

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

To prove: If f ( x ) and g ( x ) are nonzero elements of F [ x ] such that f ( x ) | g ( x ) and g ( x ) | f ( x ) then f ( x ) = a g ( x ) .

Solution Summary: The author explains that if f(x) is a nonzero polynomial in F left[x]right], then there exists

Concept explainers

Videos

To prove: If f(x) and g(x) are nonzero elements of F[x] such that f(x)|g(x) and g(x)|f(x) then f(x)=ag(x).

Want to see the full answer?

Chapter 8 Solutions