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.

We know localisation of integral domain is integral domain. So $K[X_{1},...X_{n-1}]$[$1/X_1\cdots X_{n-1}$] is an integral domain hence the ideal ($X_1\cdots X_n - 1$)`is prime and $K[X_{1},\ldots,X_{n}]$ is an UFD so ($X_1\cdots X_n - 1$) has to be irreducible . 
