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Determine the Galois groups of the following polynomials over the fields indicated:

a. x⁴ - 5 over Q; over Q(√5); over Q(√5i).

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Final answer:

The Galois group of x⁴ - 5 over ℚ is A₄, over ℚ(√5) it is C2, and over ℚ(√5i) it is C2, based on the degree of the extensions and the factorization of the polynomial in each field.

Step-by-step explanation:

To determine the Galois group of the polynomial x´ - 5 over different fields, we start with the root of the polynomial, which are the fourth roots of 5. The polynomial is irreducible over ℚ , which means the splitting field is a degree 4 extension, giving us a Galois group that is isomorphic to A₄, the alternating group on four elements.

Over ℚ(√5), the polynomial factors into two irreducible quadratics, and the splitting field is obtained by adjoining a square root of one of these quadratics, giving a C2 Galois group corresponding to a quadratic extension.

Over ℚ(√5i), all roots are obtainable by adjoining a single square root, since we already have the i to construct the four roots, so the Galois group will be C2 again for this extension.

User Rutwick Gangurde
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