Compute this Galois group – degree bounds, transitive subgroups, discriminants, and cycle types from reduction modulo primes.
The Galois correspondence – the fundamental theorem, and criteria for irreducibility, separability, normality, and Galois extensions.
Cyclotomic and radical extensions – \(\Gal(\QQ(\zeta_n)/\QQ)\cong(\ZZ/n)^\times\), splitting fields of \(x^n-a\), and solvability by radicals.
Worked examples – Galois groups of quadratics, cubics, and quartics.
Field extensions, separability, and finite fields are in Fields.