I saw the following statement "If polynomial ring $k[X_1,\cdots,X_n]$ is field then this implies that $n=0$" but I can't understand this statement. I want to completed proof of this statement.
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HINT $\rm\qquad\rm X_1 \ f\:(X_1,\ldots,X_n) = 1 \ $ in $\rm\: k[X]\:\ \Rightarrow\: \ 0 = 1 \ $ in $\rm\: k \ \ $ by evaluating at $\rm\ X_i = 0\:. $
NOTE $\:\ $ This has a very instructive universal interpretation: if $\rm\: x\:$ is a unit in $\rm\: R[x]\:$ then so too is every $\rm\: R$-algebra element $\rm\: r\:,\:$ as follows by evaluating $\ \rm x \ f(x) = 1 \ $ at $\rm\ x = r\:.\:$ Therefore to present a counterexample it suffices to exhibit any nonunit in any $\rm R$-algebra. $\:$ A natural choice is the nonunit $\;\rm 0\in R\:,\:$ which yields the above simple proof.