The task is to determine if $\mathbb R[x]$, which represents the set of all polynomials with real coefficients, is a field. My response is that it is not, since I feel that not all of these polynomials have a multiplicative inverse.
If I am correct, how would I show this in detail?
What is the inverse of the polynomial $X$?
Suppose $P(X)$ were its inverse; say $P(X) = a_n X^n + Q(X)$, where all the powers of $X$ in $Q$ are of order less than $n$, and where $a_n \not = 0$. Then $1=X P(X) = a_n X^{n+1} + X Q(X)$ where $X Q(X)$ has no $X^{n+1}$ coefficient, so the $X^{n+1}$ coefficient of $X P(X)$ cannot be $0$, so $X P(X)$ cannot be $1$.