10.2 The Main Theorem of Galois Theory
1.
a. 
is the splitting field of x5 − 1 over 
, and 
is separable as char 
. By Example 4 §10.1, 
where σ(u) = u2 (because 
. The lattices are:
is the splitting field of x5 − 1 over , and is separable as char . By Example 4 §10.1, where σ(u) = u2 (because . The lattices are:
Thus H
is the only intermediate field, 
is Galois (as H 
G), and 
. We have σ2(u) = u4 = u−1 so σ2(u + u4) = (u + u4). Hence so, since , . Of course
is the only intermediate field, is Galois (as H G), and . 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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