#15 on the book. Certainly! Here is the transcription of the text from the image:---**4.2 Divisibility in F[x]** 95(b) Give an example in ℤ[x] to show that part (a) may be false if the leading coefficient of g(x) is not a unit. [Hint: Exercise 5(b) with ℤ in place of ℚ.]15. Let R be a commutative ring with identity and a ∈ R. (a) If 1 + ax is a unit in R[x], show that 1 – a is a unit in R. [Hint: Consider 1 = ax + ...] (b) If a² = a, show that 1 + ax is a unit in R[x].16. Let R be a commutative ring with identity and a ∈ R. If 1 + ax is a unit in R[x], show that a is 0 or some integer n > 0. [Hint: Suppose that the inverse of 1 + ax is b₀ + b₁x + b₂x² + ... + bₙxⁿ. Since their product is 1₀ = 1_R (Why) and the other coefficients are all 0.]17. Let R be an integral domain. Assume that the Division Algorithm always holds in R[x]. Prove that R is a field.18. Let φ: R[x] → R be the function that maps each polynomial in R[x] onto its constant term (an element of R). Show that φ is a surjective homomorphism of rings.19. Let φ: ℤ[x] → ℤ be the function that maps the polynomial a₀ + a₁x + ... + aₙxⁿ in ℤ[x] onto the polynomial [a₀] + [a₁]x + [a₂]x² + ... + [aₙ]xⁿ, where [aᵢ] denotes the class of aᵢ modulo m. Prove that φ is a homomorphism of rings.20. Let D: R[x] → R[x] be the derivative map defined by D(a₀ + a₁x + a₂x² + ... + aₙxⁿ) = a₁ + 2a₂x + 3a₃x² +

Name: #15 on the book. Certainly! Here is the transcription of the text from
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Description: #15 on the book. Certainly! Here is the transcription of the text from the image:---**4.2 Divisibility in F[x]** 95(b) Give an example in ℤ[x] to show that part (a) may be false if the leading coefficient of g(x) is not a unit. [Hint: Exercise 5(b) w

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Math

Advanced Math

5. Let R be a commutative ring with identity and a e R. O a- O show that + ax is a unit in R[3), [HnE: Consider || – a (O) Fa'- On, show that, + axisa unit in R(3). 6. Let R be a commutative ring with identity and a e R. If l + ax is a unit in R(x), show that a - Og for some integer n>0. [Hint: Suppose that the inverse of la+ ax is by + bx + byx +...+ b. Since their product is l, bo- la (Why?) and the other coefficients are all O)

5. Let R be a commutative ring with identity and a e R. O a- O show that + ax is a unit in R[3), [HnE: Consider || – a (O) Fa'- On, show that, + axisa unit in R(3). 6. Let R be a commutative ring with identity and a e R. If l + ax is a unit in R(x), show that a - Og for some integer n>0. [Hint: Suppose that the inverse of la+ ax is by + bx + byx +...+ b. Since their product is l, bo- la (Why?) and the other coefficients are all O)

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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