
186 Abstract Rings and Fields
Since I + 1 is the identity element of Z/I, we have found a multiplicative inverse for
the given element I + r. Thus every non-zero element of Z/I has an inverse, so that
Z
n
= Z/I is a field.
From now on, when dealing with Z
n
, we revert to the usual convention and write
the elements as 0,1,2,...,n −1 rather than I,I + 1,I + 2,...,I + n −1.
16.3 Polynomials Over General Rings
We now introduce polynomials with coefficients in a given ring. The main point
to bear in mind is that identifying polynomials with functions, as we cheerfully did
in Chapter 2 for coefficients in C, is no longer a good idea, because Proposition 2.3,
which ...