(a) Is the polynomial 1+ 2r-r¹ € Z₁[r] irreducible over Z₁? Justify your answer.
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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![Question 7.
(a) Is the polynomial 1 + 2r-r² € Z₁[r] irreducible over Z₁? Justify your answer.
(b) Given two polynomials q(z) = 1-3r+r³ and g(x) = r²-5. Find a reduced form of
J + g(x) in Z₂[r]/IJ, where J = (g(x)).
(c) Given two polynomials p(x) = x² +3 and f(x) = ³r²+2. Find the multiplicative
inverse of J + f(x) in the quotient field Q[r]/I for I = (p(x)).
(d) Given the mapping : Q[r]/→ C defined by 0(+(m+nx)) = m+n√√2, vm, n € Q,
in terms of := (²+4). Prove or disprove: Q[r]/~ (G,+), where G is the image
of Q[r]/ under the mapping 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64600901-0120-4137-882a-fc73440260d5%2Fe05a006e-ad8f-40f8-8aaa-1026faab817f%2F1i04mi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 7.
(a) Is the polynomial 1 + 2r-r² € Z₁[r] irreducible over Z₁? Justify your answer.
(b) Given two polynomials q(z) = 1-3r+r³ and g(x) = r²-5. Find a reduced form of
J + g(x) in Z₂[r]/IJ, where J = (g(x)).
(c) Given two polynomials p(x) = x² +3 and f(x) = ³r²+2. Find the multiplicative
inverse of J + f(x) in the quotient field Q[r]/I for I = (p(x)).
(d) Given the mapping : Q[r]/→ C defined by 0(+(m+nx)) = m+n√√2, vm, n € Q,
in terms of := (²+4). Prove or disprove: Q[r]/~ (G,+), where G is the image
of Q[r]/ under the mapping 0.
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