Consider the mapping:

given by

.

What is the image

? Why would you expect this? What is the maximum possible degree of a "polynomial in

"? Does this have anything to do with the minimal polynomial of

(and just what is this minimal polynomial,

anyway)? What is the relationship between the maximal degree of such an

-polynomial and the dimension of

as a real vector space?

Express the complex numbers as a quotient ring of the ring of real polynomials. Which coset in the quotient ring is the pre-image of

? Explain how the multiplication in the quotient ring corresponds to the "usual" multiplication in

using the distributive laws of a field, and the fact that

.