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III. Sample Problems Consider f(x) = x4 - x2 - 2 E Q[x]. Find a splitting field K of f(x) over Q. Compute [K : Q]. Suppose F UI K is a field extension of degree p, where p is a prime number. Consider an element a E K \F. (a) Find, with proof, the degree of a over F. (b) Prove that K is a simple extension of F. Let p be a prime number. Find the splitting field of x°P - 1 over F, where: (a) F=Q. (b) F=C. Assume that the characteristic of F is 0. Prove that any degree 2 field extension of F is normal. Let G be the Galois group of x6 - 1 over Q. Prove that G is abelian and identify G (that is, write G as a cyclic group or as a product of cyclic groups).

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