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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.)

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.2: Derivatives Of Products And Quotients
Problem 35E
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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.
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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