Let F be a field. If ƒ(x) = ªo+ª₁x+...+ª₂_₁2²−¹+ªn" as f'(x) = a₁ +2a₁2 + ... + (n − 1)an-12″−2+na-¹. (a) Prove that the formal derivative satisfies the following properties: € F[¹], the formal derivative of f(x), denoted by f'(x), is defined Now, consider the set R = {f(x) ≤ F[x] : f'(1) = ƒ"(1) = 0}. (ƒ+g)'(x) = f'(x) + g'(x) and (fg)'(x) = f'(x)g(x) + f(x)g'(x), for f(x), g(x) € F[x]. (b) Show that R is a subring of F[r]. (e) Show that R is an integral domain. (In general, a subring of an integral domain need not be an integral domain.)
Let F be a field. If ƒ(x) = ªo+ª₁x+...+ª₂_₁2²−¹+ªn" as f'(x) = a₁ +2a₁2 + ... + (n − 1)an-12″−2+na-¹. (a) Prove that the formal derivative satisfies the following properties: € F[¹], the formal derivative of f(x), denoted by f'(x), is defined Now, consider the set R = {f(x) ≤ F[x] : f'(1) = ƒ"(1) = 0}. (ƒ+g)'(x) = f'(x) + g'(x) and (fg)'(x) = f'(x)g(x) + f(x)g'(x), for f(x), g(x) € F[x]. (b) Show that R is a subring of F[r]. (e) Show that R is an integral domain. (In general, a subring of an integral domain need not be an integral domain.)
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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![Let F be a field. If f(x) = ao+a₁x+...+an-1x²-¹+anxª F[r], the formal derivative of f(x), denoted by f'(x), is defined
as f'(x) = a₁ +20₁2+...+ (n − 1)an-12"-2 +nana-1. Now, consider the set R = {f(x) = F[x]: f'(1) = f(1) = 0}.
(a) Prove that the formal derivative satisfies the following properties:
(f+g)'(x) = f'(x) + g'(x) and (fg)'(x) = f'(x)g(x) + f(x)g'(x), for f(x), g(x) = F[x].
(b) Show that R is a subring of F[r].
(c) Show that R is an integral domain. (In general, a subring of an integral domain need not be an integral domain.)
(d) Show that (x - 1)³ and (x - 1)4 are irreducible elements of R.
(e) Use the result of (d) to show that R is not a UFD.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57965bde-11fe-4c54-9810-affe5cb488a2%2F060f8a1f-138a-4610-b531-f791d71c658d%2Fvegmjxq_processed.png&w=3840&q=75)
Transcribed Image Text:Let F be a field. If f(x) = ao+a₁x+...+an-1x²-¹+anxª F[r], the formal derivative of f(x), denoted by f'(x), is defined
as f'(x) = a₁ +20₁2+...+ (n − 1)an-12"-2 +nana-1. Now, consider the set R = {f(x) = F[x]: f'(1) = f(1) = 0}.
(a) Prove that the formal derivative satisfies the following properties:
(f+g)'(x) = f'(x) + g'(x) and (fg)'(x) = f'(x)g(x) + f(x)g'(x), for f(x), g(x) = F[x].
(b) Show that R is a subring of F[r].
(c) Show that R is an integral domain. (In general, a subring of an integral domain need not be an integral domain.)
(d) Show that (x - 1)³ and (x - 1)4 are irreducible elements of R.
(e) Use the result of (d) to show that R is not a UFD.
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