10.2 The Main Theorem of Galois Theory

1.
a. img is the splitting field of x5 − 1 over img, and img is separable as char img. By Example 4 §10.1, img where σ(u) = u2 (because img. The lattices are:
img
Thus Himg is the only intermediate field, img is Galois (as H img G), and img. We have σ2(u) = u4 = u−1 so σ2(u + u4) = (u + u4). Hence so, since , . Of course
c. splits (x2 + 1)(x2 − 3) over , so it is a Galois extension. Clearly ...

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