Section 4.4
Inverses and fields
In the newly constructed arithmetical systems we have not yet considered division, since it comes with various complications.
Let R be one of the rings Z, Q, R, C, or Z/mZ.
Suppose that d is a nonconstant polynomial in R[X]. The
element f 
R[X]/(d)
is called invertible
with respect to multiplication if there exists a
g
R[X]/(d) satisfying fg = 1.
Such an element g
is called an inverse of f and is denoted
by 1/f or f-1.
The set of invertible elements in S = R[X]/(d) is often denoted by S*.
If all nonzero elements are invertible, we call the ring a field. Later we will determine necessary conditions for the ring R[X]/(d) to be a field.
We have already encountered several fields, for example, Q, R, C, and Z/pZ with p prime. (See Chapter 2 for the case Z/pZ.)