Intermezzo: An Explicit Example X5 – 777X + 7

We recollect what we have developed so far on the topic of Galois theory with the help of an explicit example. It is a first application based solely of the fact that the Galois group is a group and that it can be interpreted as a subgroup in the group S(5). Without knowing the roots explicitly, we can nevertheless calculate the Galois group. In Chap. 18 we will see that this implies that the roots cannot be written as an algebraic expression with successive radicals .