True or False. Please explain. f. Let a EC be algebraic over Q of degree n. If f(a)=0 for nonzero f(x) = Q[x], then(degree f(x)) ≥ n.g. Let a EC be algebraic over Q of degree n. If f(a) = 0 for nonzero f(x) = R[x], then(degree f(x)) ≥ n.h. Every nonconstant polynomial in F[x] has a zero in some extension field of F.i. Every nonconstant polynomial in F[x] has a zero in every extension field of F.

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True or False. Please explain.

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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Question

True or False. Please explain.

f. Let a EC be algebraic over Q of degree n. If f(a)=0 for nonzero f(x) = Q[x], then
(degree f(x)) ≥ n.
g. Let a EC be algebraic over Q of degree n. If f(a) = 0 for nonzero f(x) = R[x], then
(degree f(x)) ≥ n.
h. Every nonconstant polynomial in F[x] has a zero in some extension field of F.
i. Every nonconstant polynomial in F[x] has a zero in every extension field of F.

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