Question: Use the Euclidean algorithm for polynomials to find the gcd of each pair of polynomials, over the designated field F. Then write the gcd as

Use the Euclidean algorithm for polynomials to find the gcd of each pair of polynomials, over the designated field F. Then write the gcd as s(x)f(x) +t(x)g(x) where s(x), t(x) ( F[x].
a) f(x) = x2 + x - 2, g(x) = x5 - x4 + x3 + x2 - x - 1 in Q[v]
b) f(x) = x4 + x3 + l, g(x) = x2 + x + lin Z2[x]
c) f(x) = x4 + 2x2 + 2x + 2, g(x) = 2x3 + 2x2 + x+ 1 in Z3[x]

Step by Step Solution

★★★★★

3.32 Rating (155 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a By the long division of polynomials we have x 5 x 4 x 3 x 2 x 1 x 3 ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

954-M-L-A-L-S (8638).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Prove Theorems 17.9 and 17.10. Euclidean Algorithm for Polynomials let fix), g(x) F[x] with degree f(x) Then rk(x), the last nonzero remainder, is a greatest common divisor of f(x), g(x), and is a...

Use the Euclidean algorithm to find the GCD. 36, 60

Use the Euclidean algorithm to find the GCD. 12, 45

Suppose Alice wants to send some sensitive information (credit card numbers, SSNs, corporate secrets, health records, invasion orders to start a land war in Asia ) to Bob. To prevent eavesdroppers...

(mod (4) This question deals with the finite field GF(28), which can be obtained as Z[x] x+x+x+x+1). This is the polynomial the AES cryptosystem uses, but it is not the only polynomial that gives...

Euclidean algorithm for polynomials Let f ( x ) and g ( x ) be polynomials over F for which deg ( f ( x ) ) > = deg ( g ( x ) ) > = 1 . Use the division algorithm for polynomials to compute...

1. Find an inverse of a modulo m for each of these pairs of relatively prime integers using the method followed in example below. a.) a = 4 , m = 9 b) a =19, m = 141 c) a = 55, m = 89 d) a = 89, m =...

1. Describe how to construct a field of order 8 = 23 as a quotient ring R/I (i.e., state what the appropriate ring R and ideal I are). In particular, describe what the elements of the ring R/I are...

Code the function greedy_predicator without using numpy/pandas Please include explanation of the code & the computational complexity To see the description of the function: Scroll down the...

complete the belo X a C def g f(x) v4 v5 v1 v6 v2 v3 v7 a) Does f define an isomorphism between Graph 1 and Graph 2? b ) Define a new function g (with g # f) that defines an isomorphism between Graph...

Do economists have any particular expertise at making normative arguments? In other words, they have expertise in making positive statements (ie., what will happen) about some economic policy, for...

Return to the previous problem. a. Suppose you hold an equally weighted portfolio of 100 stocks with the same alpha, beta, and residual standard deviation as Waterworks. Assume the residual returns...

If an investment account starts with $3,900, and grows with 2.1% interest, compounded annually, how much is the account worth after 15 years? Round your answer to the nearest dollar . Do NOT round...

Which one of the following will help you, the project manager, complete the needed Communications Management Plan by identifying the stakeholders and their communication needs? A. Identification of...

Data collected from 493 women are summarized in the table. a. Find the mean and standard deviation of the heights, and sketch a histogram of the data. b. Write an equation based on the model y =...

The data at right show the pulse rates of 50 people. a. Find the mean and standard deviation of the data and sketch a histogram. b. Sketch a distribution that approximates the histogram. c. Find the...

Ridge counts in fingerprints are approximately normally distributed with a mean of about 150 and a standard deviation of about 50. Find the probability that a randomly chosen individual has a ridge...

You have just completed the first year of operation for your business and have the following information: sales, $200,000; cost of goods, $140,000; rent, $18,000; utilities, $8,400; insurance,...

Derek will deposit $1,789.00 per year into an account starting today and ending in year 21.00. The account that earns 8.00%. How much will be in the account 21.0 years from today? Attempts Remaining:...

Risky Business wants to know the payback period, NPV, IRR, MIRR, and Pl of this project. The appropriate discount rate for the project is 13%. If the cutoff period is 6 years for major projects,...