Galois Theory, 5th Edition

Chapter 13Worked Examples

DOI: 10.1201/9781003213949-13

The Fundamental Theorem of Galois theory is quite a lot to take in at one go, so it is worth spending some time thinking it through. We therefore analyse how the Galois correspondence works out on some extended examples. With the machinery currently at our disposal, these examples have to be relatively simple. In particular, most of these examples use polynomials whose zeros can be calculated explicitly, either in terms of radicals or the complex exponential function. In all cases we calculate the Galois group over the rationals $\mathbb{Q}$.

13.1 Examples of Galois Groups

Example 13.1. The first example is a simple warm-up problem. Let $f(t)=t^2-5$ over $\mathbb{Q}$. The zeros are $\pm \sqrt{5}$, so the splitting field ...