The exercises in Dummit and Foote (3rd Edition) are notorious for requiring both conceptual understanding and computational skill. 1. Master the Galois Group Computation
Which from Chapter 14 you are currently trying to solve. Dummit And Foote Solutions Chapter 14
Section 14.3 and 14.5 explore special classes of extensions. The exercises in Dummit and Foote (3rd Edition)
Applies Galois theory to fields with a prime-power number of elements. Dummit And Foote Solutions Chapter 14