The Galois group
In this video, we introduce the Galois group, which captures the symmetry of a field extension. It is really the hero of Galois theory. We look at the Galois group in terms of mapping roots of irreducible polynomials to other roots, a viewpoint which allows us to give an upper bound on the size of the Galois group and thus calculate it a number of examples. A key example we give, of essential importance in number theory, is the cyclotomic field, that is, the field generated by a root of unity. We look at its Galois group in detail as the multiplicative of the ring Z/n. We finish up with applications of the Galois group to computing minimal polynomials. This gives a completely different group-theoretic perspective which reduces the problem to the study of Galois groups and orbits of roots under the Galois action.