Problem 3. Let F be a field, and let f(x) E F[x]. i. Prove that g(x) + (f(x)) has a multiplicative inverse in F[æ]/(f(x)) if and only if gcd(g(x), f(x)) = 1. ii. Prove that F[x]/(f(x)) is a field if and only if f(x) is irreducible in F[x]. (Definition: f(x) E F[x] is irreducible if and only if, whenever f(x) = a(x)b(x) with a(x), b(x) E F[x], one of a(x), b(x) must be constant.)
Problem 3. Let F be a field, and let f(x) E F[x]. i. Prove that g(x) + (f(x)) has a multiplicative inverse in F[æ]/(f(x)) if and only if gcd(g(x), f(x)) = 1. ii. Prove that F[x]/(f(x)) is a field if and only if f(x) is irreducible in F[x]. (Definition: f(x) E F[x] is irreducible if and only if, whenever f(x) = a(x)b(x) with a(x), b(x) E F[x], one of a(x), b(x) must be constant.)
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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![Problem 3. Let F be a field, and let f(x) E F[x].
i. Prove that g(x) + (f(x)) has a multiplicative inverse in F[æ]/(f(x)) if and only if gcd(g(x), f(x)) = 1.
ii. Prove that F[x]/(f(x)) is a field if and only if f(x) is irreducible in F[x]. (Definition: f(x) E F[x] is
irreducible if and only if, whenever f(x) = a(x)b(x) with a(x), b(x) E F[x], one of a(x), b(x) must be
constant.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe0df5ced-c4f7-45b3-9afb-29f734e32fc3%2F6a78e1ed-50ed-4ac3-9f64-9a5b3b613499%2F4yc2ep4_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 3. Let F be a field, and let f(x) E F[x].
i. Prove that g(x) + (f(x)) has a multiplicative inverse in F[æ]/(f(x)) if and only if gcd(g(x), f(x)) = 1.
ii. Prove that F[x]/(f(x)) is a field if and only if f(x) is irreducible in F[x]. (Definition: f(x) E F[x] is
irreducible if and only if, whenever f(x) = a(x)b(x) with a(x), b(x) E F[x], one of a(x), b(x) must be
constant.)
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