3) Differentiation D: F[t] ---> F[t], is linear and satisfies Dt¹ = nt¹-1. It satisfies the usual rules for derivatives. Let f € F[t] be monic. We write f for Df. (a) Let c E F be a root of f. With p = t - c, show that f (c) #0 iff ep(f) = 1. Suppose that F contains Q, the rational numbers. (b) Show that gcd (f, f) = 1 iff Iff f = rad(f), ep(f) ≤1 i.e. V primes p of F[t]
3) Differentiation D: F[t] ---> F[t], is linear and satisfies Dt¹ = nt¹-1. It satisfies the usual rules for derivatives. Let f € F[t] be monic. We write f for Df. (a) Let c E F be a root of f. With p = t - c, show that f (c) #0 iff ep(f) = 1. Suppose that F contains Q, the rational numbers. (b) Show that gcd (f, f) = 1 iff Iff f = rad(f), ep(f) ≤1 i.e. V primes p of F[t]
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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nt-1. It satisfies the
3) Differentiation D: F[t] - -> F[t], is linear and satisfies Dt¹
usual rules for derivatives. Let f € F[t] be monic. We write f for Df.
(a) Let c E F be a root of f. With p = t - c, show that f(c) #0 iff ep(f) = 1.
Suppose that F contains Q, the rational numbers.
(b) Show that gcd (f, f) = 1
iff
Iff
f = rad(f),
ep(f) ≤ 1
i.e.
V primes p of F[t]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F882ce437-6984-48a2-b1c1-0c70a26630a5%2F01400dfa-549e-4757-a9c4-d5ed8d9db9cc%2Ff4vxqmm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:=
nt-1. It satisfies the
3) Differentiation D: F[t] - -> F[t], is linear and satisfies Dt¹
usual rules for derivatives. Let f € F[t] be monic. We write f for Df.
(a) Let c E F be a root of f. With p = t - c, show that f(c) #0 iff ep(f) = 1.
Suppose that F contains Q, the rational numbers.
(b) Show that gcd (f, f) = 1
iff
Iff
f = rad(f),
ep(f) ≤ 1
i.e.
V primes p of F[t]
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