Describe the Galois group, all its subgroups, and the corresponding fixed fields, for the splitting fields...

 

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Describe the Galois group, all its subgroups, and the corresponding fixed fields, for the splitting fields of the following polynomials: (a) x² + x + 1 € Q[r]; (b) 2 - 1 € Q[r]; (c) (2²-2)(x² - 3)(x² - 5)(r² - 7) = Q[¹]. Describe the Galois group, all its subgroups, and the corresponding fixed fields, for the splitting fields of the following polynomials: (a) x² + x + 1 € Q[r]; (b) 2 - 1 € Q[r]; (c) (2²-2)(x² - 3)(x² - 5)(r² - 7) = Q[¹].

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Answer a The polynomial 2 x 1 over Q has roots r1 12 i72 and r2 12 i72 The splitting field of this polynomial over Q is the field F Qr1 r2 obtained by ... View the full answer