We should notice that in general algebraic theory, there is no reason why the algebras on the diagonal, that is, should be commutative. This is the reason that we use the square brackets in the notation. The general, not necessarily commutative, polynomial algebra over R with M variables should
then be named R.
For the rest of this text, we will let R = k, an algebraically closed field of characteristic 0. And of course, because we are giving the physical interpretation, we think of k as the field of complex numbers . Read more