Invitation to Galois Theory

We will sketch, in a rather informal way, two historically important successes of Galois theory. In this invitation, we will only use the well-known fact that the data of a subfield k of K equips K with the structure of a k-vector space. The dimension, finite or not, is denoted \([K:k]\) and is also called the degree of the extension \(K/k\) .