Definition 17.8.1 (Galois extension and Galois group). A finite extension is Galois if it is both normal and separable. The Galois group is the group of field automorphisms of fixing pointwise. For a Galois extension, .
Example 17.8.1 (Galois groups of quadratic extensions). Let be a non-square. The extension is Galois of degree 2. The Galois group is where . So .
Example 17.8.2 (Splitting field of ). The splitting field of over is where . The roots are . We have (since and ). The Galois group is : it acts faithfully on the three roots, and .