Q7 Define f(x)=³-3x+1, and o(a)=x²-2. The following information will be useful: f is irreducible over Q. • f(o(x)) = g(x)f(x) where g(x)=3-3x - 1. (You do not need to check either of these things although both are straightforward calculations.) (a) How many distinct roots does f have in C? Explain your answer. (b) Show that maps any root of f to another, distinct, root of f. I.e., if a is a root of f, then o(a) is a root of f, and moreover o(a) # a.
Q7 Define f(x)=³-3x+1, and o(a)=x²-2. The following information will be useful: f is irreducible over Q. • f(o(x)) = g(x)f(x) where g(x)=3-3x - 1. (You do not need to check either of these things although both are straightforward calculations.) (a) How many distinct roots does f have in C? Explain your answer. (b) Show that maps any root of f to another, distinct, root of f. I.e., if a is a root of f, then o(a) is a root of f, and moreover o(a) # a.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 12E: a. Find a nonconstant polynomial in Z4[ x ], if one exists, that is a unit. b. Find a nonconstant...
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