Another great set of notes by H.W.Lenstra which discusses algebra in general, and in particular Field and Galois theory at the end, can be found here (pdf). Two quotes about the approach:
We indulge next in a casual and motivational comparison of the classical and modern approaches to Galois theory. In all current textbooks, Galois theory is studied using ﬁnite separable ﬁeld extensions L of a given base ﬁeld K. Our approach follows that of the Grothendieck formulation, in which the objects under consideration are ﬁnite étale K-algebras A. We now consider the relation between the two perspectives.
One can track Galois theory through the years from being a discussion of polynomials, to an exploration of splitting ﬁelds, and ﬁnally to the Grothendieck formulation that we have used in this unit.