(a) Use the Factor Theorem to construct a single polynomial f(x) € Z5[r] such that every element of Z5 is a root of f(x). (b) Find all odd primes p for which z + 2 is a factor of r-r³+r²-x in Z. (c) For which k EC is a +i a factor of ir +3r7+x6 - 2ix + ke C[x]? (c) Find all of the associates of 2-3x² + 3x6 € Z5[r]. (d) Prove that Z[V-5], together with the operations of addition + and multiplication, constitute a subring of C, +, >. Is the integer 2 + √5 reducible or irreducible in ZV-5]? Justify your answer.
(a) Use the Factor Theorem to construct a single polynomial f(x) € Z5[r] such that every element of Z5 is a root of f(x). (b) Find all odd primes p for which z + 2 is a factor of r-r³+r²-x in Z. (c) For which k EC is a +i a factor of ir +3r7+x6 - 2ix + ke C[x]? (c) Find all of the associates of 2-3x² + 3x6 € Z5[r]. (d) Prove that Z[V-5], together with the operations of addition + and multiplication, constitute a subring of C, +, >. Is the integer 2 + √5 reducible or irreducible in ZV-5]? 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
![then R[T] ~ S[T].
Question 6.
(a) Use the Factor Theorem to construct a single polynomial f(x) € Z5[r] such that every
element of Z5 is a root of f(r).
(b) Find all odd primes p for which z + 2 is a factor of r¹-³+²-r in Zp.
(c) For which k EC is z+i a factor of irº+3x² + x³ = 2ix + k € C[r]?
I
(c) Find all of the associates of 2-3x² + 3x6 € Zs[r].
(d) Prove that Z[√-5], together with the operations of addition + and multiplication,
constitute a subring of C, +, >. Is the integer 2 √ 5 reducible or irreducible in
+
Z√5]? Justify your answer.
Question 7.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40a2e609-fa1b-4aa0-bf1d-f78d8a938941%2Fb4f0d733-da7a-4fea-8719-0ceb5056cac4%2F4qwjqe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:then R[T] ~ S[T].
Question 6.
(a) Use the Factor Theorem to construct a single polynomial f(x) € Z5[r] such that every
element of Z5 is a root of f(r).
(b) Find all odd primes p for which z + 2 is a factor of r¹-³+²-r in Zp.
(c) For which k EC is z+i a factor of irº+3x² + x³ = 2ix + k € C[r]?
I
(c) Find all of the associates of 2-3x² + 3x6 € Zs[r].
(d) Prove that Z[√-5], together with the operations of addition + and multiplication,
constitute a subring of C, +, >. Is the integer 2 √ 5 reducible or irreducible in
+
Z√5]? Justify your answer.
Question 7.
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