Prove if K and E be the extension of a field F with [K:F] finite and assume that both K and E are subfields of some larger field. If K is a Galois extension of F then KE is a Glois extension of E and G(KE, E)≅G(K,K∩E).

Prove if K and E be the extension of a field F with [K:F] finite and assume that both K and E are subfields of some larger field. If K is a Galois extension of F then KE is a Glois extension of E and G(KE, E)≅G(K,K∩E).

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 18E: Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily...

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Question

Prove if K and E be the extension of a field F with

[K:F] finite and assume that both K and E are
subfields of some larger field. If K is a Galois
extension of F then KE is a Glois extension of
E and G(KE, E)≅G(K,K∩E).

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