# Why can’t the Polynomial Ring be a Field?

I’m currently studying Polynomial Rings, but I can’t figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This implies an inverse multiple for every Element in the Set.

The book doesn’t elaborate on this, however. I don’t understand why a Polynomial Ring couldn’t have an inverse multiplicative for every element (at least in the Whole numbers, and it’s already given that it has a neutral element). Could somebody please explain why this can’t be so?

Hint $\rm\quad\rm x \, f(x) = 1 \,$ in $\,\rm R[x]\ \Rightarrow \ 0 = 1 \,$ in $\,\rm R, \,$ by evaluating at $\rm\ x = 0$
Remark $\$ This has a very instructive universal interpretation: if $\rm\, x\,$ is a unit in $\rm\, R[x]\,$ then so too is every $\rm\, R$-algebra element $\rm\, r,\,$ as follows by evaluating $\ \rm x \ f(x) = 1 \$ at $\rm\ x = r\,.\,$ Therefore to present a counterexample it suffices to exhibit any nonunit in any $\rm R$-algebra.  A natural choice is the nonunit $\,\rm 0\in R,\,$ which yields the above proof.