Irreducible element of the ring. Element $X_1 X_2 \cdots X_n - 1$ is irreducible in $K[X_{1},\ldots,X_{n}]$ for $n\ge 1$, where $K$
is a field.
For $n=2,3$ it is easy to see that the element is irreducible but for higher value of $n$ I was trying to use inductive argument but not getting any idea.

Hint $\, $ It's a special case of: $\rm\:ax\!+\!b\in D[x]\:$ is irreducible for domain $\rm\:D\:$ and $\rm\:0\ne a\:$ and $\rm\,b\,$coprime. Indeed, if reducible then comparing degrees (using $\rm\,D\,$ domain) we infer that one factor would be a constant nonunit, so it would be a nonunit common divisor of $\rm\,a,b,\,$ contra hypothesis. This may fail if $\rm\,D\,$ is not a domain, e.g. $\rm\,(2x\!+\!1)(3x\!-\!1) = x\!-\!1 \in \Bbb Z/6[x].$
Yours is the special case $\rm\:D = K[X_1,\ldots,X_{n-1}],\ a = X_1\cdots X_{n-1},\ x = X_n,\ b = -1.$
Remark $\ $ This "primitivity" criterion for irreducibility generalizes to higher degree polynomials. Namely, if $\rm\,D\,$ is a domain with fraction field $\rm\,K\,$ then, for a nonconstant polynomial $\rm\,f\in D[x]$
$$\rm f\,\ is\ prime\ in\ D[x]\iff  f\,\ is\ prime (= irreducible)\ in\ K[x]\ and\,\ f\,\ is\ superprimitive $$
$$\rm where\,\ f\,\ is\ {\bf superprimitive}\ in\ D[x]\,\ :=\,\  d\,|\,cf\, \Rightarrow\, d\,|\,c\,\ \ for\ all\,\ c,d\in D^*$$
