Answered: K(g^(1/2))/K

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K(g^(1/2))/K

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 24E: If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type...

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Question

Say we have a field K and g, an element of K, where there is no solution to the equation x^2=g for x an element of K. Is the field extension K(g^(1/2))/K necessarily Galois?

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Step 1: the field extension K(g^(1/2))/K is not necessarily Galois.

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Step 2: Example, let k be the field of rational functions in two varibles.

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