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?

Answer

Hint xf(x)=1 in R[x]  0=1 in R, by evaluating at  x=0

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

Attribution
Source : Link , Question Author : IAE , Answer Author : Bill Dubuque

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