Consider f(x) = x³ + 2x² + 3, g(x) = x³ + 2x² + 4 in the polynomial ring F5[2]. Denote F125 = F5[x]/(f(x)) and R125 = F5[x]/(g(x)). 1. Since f(x) has degree 3, it is irreducible if and only if it has no roots. Verify this by computing the values of f on the 5 elements of F5. This shows that F125 is a field. f(0) = 3 . f(1) = 1 f(2)= 1 f(3) = 4 f(4) = 0 2. Find all the roots of g(x). Give your answer as a list of numbers separated by commas. Roots of g(x)= 2 Since there is at least one, g(z) is reducible and so R125 is not a field. 3. In F125, compute (2x² + x +1)(3x² + 2x + 2) = x^4+2x^3+4x^2+4x+2 with coefficients from F5. 4. 3. In R125, compute (2x² + x + 1)(3x² + 2x + 2) = less with coefficients from F5. 5. In F125, compute (2x² + x + 1)¬¹ ; coefficients from F5. giving your answer as polynomial of degree 2 or less giving your answer as polynomial of degree 2 or giving your answer as polynomial of degree 2 or less with
Consider f(x) = x³ + 2x² + 3, g(x) = x³ + 2x² + 4 in the polynomial ring F5[2]. Denote F125 = F5[x]/(f(x)) and R125 = F5[x]/(g(x)). 1. Since f(x) has degree 3, it is irreducible if and only if it has no roots. Verify this by computing the values of f on the 5 elements of F5. This shows that F125 is a field. f(0) = 3 . f(1) = 1 f(2)= 1 f(3) = 4 f(4) = 0 2. Find all the roots of g(x). Give your answer as a list of numbers separated by commas. Roots of g(x)= 2 Since there is at least one, g(z) is reducible and so R125 is not a field. 3. In F125, compute (2x² + x +1)(3x² + 2x + 2) = x^4+2x^3+4x^2+4x+2 with coefficients from F5. 4. 3. In R125, compute (2x² + x + 1)(3x² + 2x + 2) = less with coefficients from F5. 5. In F125, compute (2x² + x + 1)¬¹ ; coefficients from F5. giving your answer as polynomial of degree 2 or less giving your answer as polynomial of degree 2 or giving your answer as polynomial of degree 2 or less with
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Consider f(x) = x³ + 2x² +3, g(x) = x³ + 2x² +4 in the polynomial ring F5[*]. Denote F125 = F5[x]/(f(x)) and R125 = F5[x]/(g(x)).
1. Since f(x) has degree 3, it is irreducible if and only if it has no roots. Verify this by computing the values of f on the 5 elements of F5. This
shows that IF125 is a field.
f(0) = 3
f(1) = 1
. f(4) = 0
ƒ(2)
= 1
5. In F125, compute (2x² + x + 1)−¹:
coefficients from F5.
f(3) =
2. Find all the roots of g(x). Give your answer as a list of numbers separated by commas.
Roots of g(x) = 2
Since there is at least one, g(x) is reducible and so R125 is not a field.
4. 3. In R125, compute (2x² + x + 1)(3x² + 2x +2):
=
less with coefficients from IF5.
= 4
3. In F125, compute (2x² + x + 1)(3x²+2x + 2) = x^4+2x^3+4x^2+4x+2
with coefficients from F5.
giving your answer as polynomial of degree 2 or less
giving your answer as polynomial of degree 2 or
giving your answer as polynomial of degree 2 or less with](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e903c09-c796-4754-aa19-01a5a83c6f2c%2F243995e7-102f-4b12-80be-dee29da6fddd%2Fwwr0zxr_processed.png&w=3840&q=75)
Transcribed Image Text:Consider f(x) = x³ + 2x² +3, g(x) = x³ + 2x² +4 in the polynomial ring F5[*]. Denote F125 = F5[x]/(f(x)) and R125 = F5[x]/(g(x)).
1. Since f(x) has degree 3, it is irreducible if and only if it has no roots. Verify this by computing the values of f on the 5 elements of F5. This
shows that IF125 is a field.
f(0) = 3
f(1) = 1
. f(4) = 0
ƒ(2)
= 1
5. In F125, compute (2x² + x + 1)−¹:
coefficients from F5.
f(3) =
2. Find all the roots of g(x). Give your answer as a list of numbers separated by commas.
Roots of g(x) = 2
Since there is at least one, g(x) is reducible and so R125 is not a field.
4. 3. In R125, compute (2x² + x + 1)(3x² + 2x +2):
=
less with coefficients from IF5.
= 4
3. In F125, compute (2x² + x + 1)(3x²+2x + 2) = x^4+2x^3+4x^2+4x+2
with coefficients from F5.
giving your answer as polynomial of degree 2 or less
giving your answer as polynomial of degree 2 or
giving your answer as polynomial of degree 2 or less with
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VIEWStep 2: Find the values of f(x) at x=0,1,2,3,4.
VIEWStep 3: Find the roots of g(x).
VIEWStep 4: Find the product of the polynomial on F of 125.
VIEWStep 5: Find the product of the polynomial on R of 125.
VIEWStep 6: Find the inverse of the given polynomial over the F of 125.
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