Let R be a field. Is R[x] a field then?

Question

Let R be a field. Is R[x] a field then? I guess the answer is NO, but it doesn't come any counterexample in my mind. thanks for your help.

Answer 1

No. For degree reasons, a polynomial $P(X)$ with positive degree cannot have an inverse in $R[X]$. This means the only invertible elements in $R[X]$ are the non-zero constants.

However it is a P.I.D., and as such, has a field of fractions, denoted $R(X)$.