ABSTRACT
Our next chapters investigate a few topics in algebra. Recall that algebra is the study of algebraic structures, i.e., sets with operations on them. We already introduced, and constructed, some elementary examples of algebraic structures such as rings and, in particular, fields. With rings/fields at our disposal one can study some other fundamental algebraic objects such as groups, vector spaces, polynomials. In what follows we briefly survey some of these. We begin with groups. In some sense groups are more fundamental than rings and fields; but in order to be able to look at more interesting examples we found it convenient to postpone the discussion of groups until this point. Groups appeared in mathematics in the context of symmetries of roots of polynomial equations; cf. the work of Galois that involved finite groups. Galois’ work inspired Lie who investigated differential equations in place of polynomial equations; this led to (continuous) Lie groups, in particular groups of matrices. Groups eventually penetrated most of mathematics and physics (Klein, Poincare´, Einstein, Cartan, Weyl).
