а) Check whether or not Q(/5)- Q(/7). b) Check whether or not if L/K and K/F are Galois extensions, then L/F is a Galois extension. Give an example, with justification, of a reducible separable polynomial over F25. c) d) Find the splitting field of x° - 4, and its degree, over Q.

а) Check whether or not Q(/5)- Q(/7). b) Check whether or not if L/K and K/F are Galois extensions, then L/F is a Galois extension. Give an example, with justification, of a reducible separable polynomial over F25. c) d) Find the splitting field of x° - 4, and its degree, over Q.

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 6TFE

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Question

Please answer all questions

6.
а)
Check whether or not Q(/5)= Q(/7).
b)
Check whether or not if L/K and K/F are Galois extensions, then L/F is a
Galois extension.
c)
Give an example, with justification, of a reducible separable polynomial
over F125.
6.
d)
Find the splitting field of x° - 4, and its degree, over Q.
7.
i)
Show that Q(2/4,i)/Q is a Galois extension.
ii)
Find the structure of its Galois group, G.

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