Let E/F be a field extension. Let a € E be algebraic of degree n over F. Let f(X) e F[X] be a non-constant polynomial of degree d. Prove that f(a) has degree at leastover F.

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ƒ(X) e F[X] be a non-constant polynomial of degree d. Prove that f(a) has degree at least over F.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 9E: Let be a field. Prove that if is a zero of then is a zero of

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Let E/F be a field extension. Let a € E be algebraic of degree n over F. Let

f(X) e F[X] be a non-constant polynomial of degree d. Prove that f(a) has degree at least
over F.

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