Р(1)p(2)p(3)mation, and find ker (T). Recall that P3 is the space of all polynomials of degree at most8. Let T : P3 → R³ be defined by T (p(x))Show that T is a linear transfor-3, P3 := {p(x) = ao + a1x + a2² + azx*| a1, a2, a3, a4 E R}.

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Answered: p(1) Р(2) p(3) mation, and find ker…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.6: Algebraic Extensions Of A Field

Problem 7E

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1. Find a monic polynomial of least degree over that has the given numbers as zeros, and a monic polynomial of least degree with real coefficients that has the given numbers as zeros. a. b. c. d. e. f. g. and h. and
a. Find a nonconstant polynomial in Z4[ x ], if one exists, that is a unit. b. Find a nonconstant polynomial in Z3[ x ], if one exists, that is a unit. c. Prove or disprove that there exist nonconstant polynomials in Zp[ x ] that are units if p is prime.
Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].

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p(1) Р(2) p(3) mation, and find ker (T). Recall that P3 is the space of all polynomials of degree at most Let T : P3 → R³ be defined by T (p(x)) = Show that T is a linear transfor- 3, P3 := {p(x) = ao + a1x + a2x² + azx*| a1, a2, a3, a4 € R}.