Definition Schwarz Function : Which sense to give to a polynomial of R^d ? and relation with multi-indexes generalization

Question

In most of definitions , but we can take the example of Wikipedia EN of Schwartz space, people use the terminology of "monomial" or "polynomial" (depending of if we take a sum or just the sup) to refer to the "$x^\alpha$" term , but i don't understand how this terminology can be used. R^d(d>1) is not even a field , how can we talk about a polynomial or monomial for terms of R^d ? I looked for the meaning of the multi-index notation $x^\alpha$ ,I get it , but that doesn't relate to a "polynomial" to me. Can someone explain , or ask me to be more clear if you don't get the question,