Let K be a field of characteristic not equal to 2. Let c E K be...

 

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Let K be a field of characteristic not equal to 2. Let c E K be any element which is not a square, and let √e be a square root in some field extension. For a, b eK with not both a, b = 0, set a = a +b√c. Set EK(√e, √a). Show that the following are equivalent: (a) E is Galois over K; (b) E = K(√e, √a'), where a' = a - b√c (c) Either an' or can' is a square in K. Show that if E is Galois over K. its Galois group is cyclic of order 4 if and only if cao' is a square in K. phic to the direct product of two cyclic groups both of whose orders are greater than 1. Let K be a field of characteristic not equal to 2. Let c E K be any element which is not a square, and let √e be a square root in some field extension. For a, b eK with not both a, b = 0, set a = a +b√c. Set EK(√e, √a). Show that the following are equivalent: (a) E is Galois over K; (b) E = K(√e, √a'), where a' = a - b√c (c) Either an' or can' is a square in K. Show that if E is Galois over K. its Galois group is cyclic of order 4 if and only if cao' is a square in K. phic to the direct product of two cyclic groups both of whose orders are greater than 1.

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