Let K, L, and N be fields such that L/K and N/K are both finiteextensions, N/K is normal and L n N = K. Let a € N\ K andlet f(r) be the minimal polynomial of a over K. Prove that f(r) isirreducible over L. (Hint: use the previous problem.)

This question is about application of normed vector space and vector calculas

Let K, L, and N be fields such that L/K and N/K are both finite extensions, N/K is normal and Ln N = K. Let a e N \K and let f(r) be the minimal polynomial of a over K. Prove that f(x) is irreducible over L. (Hint: use the previous problem.)

Let K, L, and N be fields such that L/K and N/K are both finite extensions, N/K is normal and Ln N = K. Let a e N \K and let f(r) be the minimal polynomial of a over K. Prove that f(x) is irreducible over L. (Hint: use the previous problem.)

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 8E: Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero ...

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Question

Let K, L, and N be fields such that L/K and N/K are both finite
extensions, N/K is normal and L n N = K. Let a € N\ K and
let f(r) be the minimal polynomial of a over K. Prove that f(r) is
irreducible over L. (Hint: use the previous problem.)

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