what is invertible elements of polynomial ring $k[x_1,...,x_n]$ ? for case $n=1$ we have $k[x_1]$ and $$ p(x)=a_{0}+a_{1} x+\ldots a_{n} x^{n} $$ then $p(x)$ is invertible element such that $a_{0}$ is a unit in $R,$ and the remaining coefficients $a_{1}, \ldots, a_{n}$ are nilpotent elements. ( $k$ is a field)
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1How many nilpotents does a field have? – Randall Feb 07 '21 at 14:56
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If $k$ is a field then for nonzero polynomials in $k[x_1]$, say $p$ and $q$ You have $$\deg(pq)=\deg(p)+\deg(q)$$ and since $\deg(1)=0$, where $1$ is the constant polynomial with coefficient $1$, You see that the image of $k^*=k\backslash\{0\}$ in $k[x_1]$ is the set of units in $k[x_1].$
Peter Melech
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