Answered: Every nonconstant polynomial in F[x]…

Every nonconstant polynomial in F[x] has a root in every extension field of F, that is if K is an extension of F and f is a nonconstant polynomial in F[x], then f has a root in K.

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 2TFE

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Question

True or False

1. Every nonconstant polynomial in F[x] has a root in every extension field of F, that is if K is an extension of F and f is a nonconstant polynomial in F[x], then f has a root in K.
2. There is no proper intermediate field of ℝ and ℂ, i.e. there exists no field E such that ℝ ≤ E ≤ ℂ but ℝ ≠ E, E ≠ ℂ.
3. If [K:F]=p for some prime p and F ≤ E ≤ F, then F=E.

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