If $F$ is a field, show that $F[x]$ is never a field.

Question

I am not sure how to go about this. Could someone please check my work and see if I am doing something wrong?

If $F$ is a field, show that $F[x]$ is never a field.

Let $p(x)=x\in F[x]$. Then if $F[x]$ is a field, $g(x)=x^{-1}\in F[x]$ such that $p(x)$ has a multiplicative inverse where $p(x)g(x)=xx^{-1}=1\in F[x]$. However, $g(x)$ is not a polynomial, and so $p(x)$ is not unit. Therefore, $F[x]$ cannot be a field.

Answer 1

If $F[x]/(x)\cong F$ is a field and $F[x]$ is a field then $x$ has a multiplicative inverse so $(x)=F[x]$. But this forces $F=\{0\}$, which is not a field.

Answer 2

Actually, in principle, it could be the case that $x^{-1}$ equals some polynomial. So, you have to check that given $p(x)=x\in F[x]$, there's no polynomial $q(x)\in F[x]$ such that $p(x)q(x)=1$.

I think you'll be able to do the details using the properties of the product and sum of polynomials with respect to the degree of the polynomials involved.