10.2 The Main Theorem of Galois Theory

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:
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 ...

Get Introduction to Abstract Algebra, Solutions Manual, 4th Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.