(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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Chapter2: Second-order Linear Odes
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answer d and e

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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